A writer, inventor, scientist, social engineer, musician,  philosopher, economist, diplomat and revolutionary, Franklin helped shape the American colonies destiny.
   Ben Franklin has the distinction of many "firsts":   He invented the first lightning rod, Franklin stove, bifocals, catheter, swimfins and odometer, as well as the first armonica. He started the first public library, the first volunteer fire company, American fire insurance company, the first hospital and was the first postmaster general.
   He founded the American Philosophical Society, drew up and was president of The Academy and College of Philadelphia (later to become the University of Pennsylvania.)

When thinking of Franklin as a chess player, Prof. George Allen offers this caveat:

Now this clever Yankee [Franklin], so economical of time in all other respects, had a perfect passion for playing Chess; and he gives no hint of ever being at a loss for Philadelphians to play with. - in site of the various attempts of a certain Chess editor to make it out otherwise - is a fair inference from the fact, that he found his match in an English woman, and had to accept the Knight from a French woman; and that some of his antagonists were strong players, who beat him soundly and easily, is rendered in the highest degree probable by the fact, that the line of hereditary Chess-talent, in one known contemporary instance, can be traced back, to the generation in question. It was, however, neither his weakness, nor their strength, in Chess, that attracted of repelled the attention of the disdainful Muses. But it happened that our Yankee friend took it into his head, one day, to perform the unaccountable feat of flying a kite at a thunder-cloud, and afterwards to dabble in rebel politics; and now, behold! a godly heap of octavos, by the biographer of Washington, lies solidly and heavily upon his memory; one entire section of this very  chapter of  THE BOOK has been devoted to his glory as a "Chess-player" forsooth; and thus he has come to have nearly as good a chance for immortality as Philidor himself; while not even the name of those who really deserve to be remembered - the men who gave him, or could have given him, "Pawn and two" at the least - has escaped the cruel god that eats up his own children: "Can haughty time be just?"

So Franklin should be remembered more for his passion for chess rather than for his skill.

While it's known the Franklin played chess in America, he undoubtedly found more chess partners abroad. In fact, contradicting what Prof. Allen mentions about Franklin finding no lack of opponents in Philadelphia, in a letter to William Straham June 20, 1752, Franklin wrote:

"...Honest David Martin, Rector of our Academy, my principal Antagonist at Chess, is dead, and the few remaining Players here are very indifferent, so that I have now no need of Stamma's 12s Pamphlet, and am glad you did not send it."

David Martin (1696-1751) was a fellow member of Franklin's American Philosophical Society. He had served as Sheriff of Hunterdon County in New Jersey for 13 or 14 years, operated a ferry across the Delaware River near  Easton, Pennsylvania and served as Marshall of Trenton, N. J.,  He moved to Philadelphia where he met Franklin. From 1849 until his death in 1851, he served as First Rector and Professor of Greek and Latin for the Academy and College of Philadelphia.

Franklin's references to chess in his autobiography are slim, but in one notable instance he wrote:

I had begun in 1733 to study languages; I soon made myself so much a master of the  French as to be able to read the books with ease. I then undertook the Italian. An acquaintance, who was also learning it, us'd often to tempt me to play chess with him. Finding this took up too much of the time I had to spare for study, I at length refus'd to play any more, unless on this condition, that the victor in every game should have a right to impose a task, either in parts of the grammar to be got by heart, or in translations, etc., which tasks the vanquish'd was to perform upon honour, before our next meeting. As we play'd pretty equally, we thus beat one another into that language. I afterwards with a little painstaking, acquir'd as much of the Spanish as to read their books also.

Franklin isn't recorded as having played much chess in England where he lived from 1757-62 and 1764-67. But there is one instance of Franklin using chess as a front for his political intrigue.

Quoting from Walter Isaacson's  Benjamin Franklin: an American Life

...[Franklin received] a curious invitation from a well-connected society matron who let it be known that she wanted to play chess with Franklin. The woman in question was Caroline Howe, the sister of Adm. Richard Howe and Gen. William Howe. They would eventually
become commanders of England's naval and land forces during the Revolution, but at the time they were both somewhat sympathetic to the American cause.
...On Christmas day, Franklin visited Mrs. Howe for another chess match. As soon as he arrived, she mentioned that her brother, Admiral Lord Richard Howe, wanted to meet him. "Will you give me leave to send for him?" she asked. Franklin readily agreed, and soon he was listening as Lord Howe showered him with compliments. "No man could ever do more towards reconciling our differences," the admiral told him. He asked Franklin to offer some suggestions, which he would communicate to the proper ministers. Franklin, wary of being caught in the middle, noted the Continental Congress had made clear what the colonies wanted. But agreed to another secret session a week later, again under the guise of visiting Mrs. Howe to play chess.

In an other quote from the same book:

In the meantime, Franklin had the pleasure of settling back into the life he loved in London. Sir John Pringle, the distinguished physician had become his best friend. The played chess, made the rounds of their regular coffeehouse clubs, and soon got in the habit of taking  summer trips together. The great Samuel Johnson biographer, James Boswell, was another acquaintance. After dropping in on one of their chess games, Boswell noted in his journal that Pringle had a "peculiar sour manner, "but that Franklin was as always, "all jollity and pleasantry."

Franklin spent time in France (accompanied by his two grandsons, sixteen-year-old William Temple Franklin and seven-year-old Benjamin Franklin Bache) between the years 1767 and 1785. He was appointed Minister to France in 1779. He became a social lion and was particularly popular with the Parisienne ladies, There he also met, and became infatuated with, Madame Brillon de Jouy, née Anne-Louise Boivin d'Hardancourt (1744-1824), called by one music critic "one of the greatest lady-players on the harpsichord in Europe." A lady of many talents, her skill with with charcoal and watercolors was also highly praised. In spite of her dissatisfaction with her husband and of Franklin's obvious attraction, their relationship was strictly platonic - mostly due to Franklin's (unsuccessful) desire that his grandson, William, should marry Madame Brillon's daughter, Cunegonde. But their relationship was very coquettish and chess-centric.

Quoting from the same book:

Madame Anne-Louise Brillon de Jouy, who was 33 when she met Franklin, was buffeted by conflicting passions and variable moods. Her husband, twenty-four years her senior (but fourteen years younger than Franklin) was wealthy, doting and unfaithful. Although she  spoke no English, she and Franklin exchanged more than 130 letters during their eight-year relationship, and she was able not only to enchant him but also to manipulate him.
They flirted over the chessboard. "She is still a little miffed," Madame Brillon teasingly wrote of herself, "about the six games of chess he won so inhumanely and she warns him she will spare nothing to get her revenge."
By March 1778, after months of just music and chess, Franklin was ready for something more. Franklin in a coy exchange told her that there were two others [of the 10 commandments] that should be included: to multiply and fill the earth, and to love one another. He had always obeyed those two very well and, he argued, should that not
"compensate for my having so often failed to respect one of the ten? I mean the one which forbids us to covet thy neighbor's wife, a commandment which (I confess) I have constantly  violated."

and later

Throughout his remaining years in France, and even in letters after his return to America, Franklin would stay emotionally attached to Madame Brillon. Their new arrangement still allowed him such liberties as playing chess with a mutual friend, late into the night, in her bathroom, while she soaked in her tub and watched. But it was, as far as bathtub chess games go, rather innocent; the tub was covered, as was the style, by a wooden plank. "I'm afraid that we may have made you very uncomfortable by keeping you so long in the bath,"  he apologized the next day, adding a wry little promise: "Never again will I consent to start a chess game with the neighbor in your bathing room. Can you forgive this indiscretion?"

According to Benjamin Franklin Book of Recipes by Hilaire Dubourcq - another letter from Katherine French writing to Franklin June 17, 1771 tells us:

Mrs. French understands that Docr. Franklin dines with the Bishop of St. Asaph's to morrow hopes he will do her the favor of dining with her on Wednesday or Thursday, both days will be giving her double pleasure, she has provided chess players for each day.

According to the The Diary and Autobiography of John Adams by Lyman H. Butterfield

Adams complains in his diary (27 May 1778) that after observing Franklin's dissolute ways he decided that "the Business of our Commission would never be done, unless I did it." Franklin, he asserts, rose late and breakfasted later, occupied his day by entertaining slavish admirers in one or another Parisian salon, and dined out almost every night. Afterward he played at games [mainly chess] or listened in company to music, and "came home at all hours from Nine to twelve O Clock at night." Given this schedule, Adams claims, it was impossible to catch Franklin except to secure an occasional quick signature. Such an easy schedule, Adams caustically concludes, was agreeable to Franklin, and no doubt contributed to his health and long life. As for Adams, he conscientiously (and rather priggishly) declined all dinner invitations, stayed at home, and occupied himself with commission business or serious reading.

While in France, Franklin very likely played a game against the Turk, von Kempelen's automaton.

The "speaking figure" mention by Franklin's grandson, was, as Mr. Valltravers of Vienna wrote in a Dec. 24, 1782  letter carried by Kempelen to introduce him to Franklin, "the figure of a child uttering the first articulate sounds of elocution. Of these I have heard it pronounce distinctly upwards of thirty words and phrases. There remain but five or six letters of the alphabet, the expression of which he intends to complete in Paris."

While no record of any game exists between Franklin and Wolfgang Von Kempelen's invention, he undoubtedly did make his appointment with Kempelen. Philidor's well known Saxony opponent, Hans, Count von Bruhl , wrote a reply to a letter from Franklin which said:

Sir: ---- I was very much flattered with the letter I had the pleasure to receive from your Excellency by means of the ingenious M. de Kempel's arrival in this country. The favorable opinion you entertain of his talent is alone sufficient to convince me of their extent and usefulness. I cannot find words to express the gratitude I feel for the honor of your remembrance. I shall, therefore, only beg leave to assure you, that it will be the pride of my life to have been noticed by one of the most distinguished characters of the age, and I shall endeavor, upon all occasions, to contribute my mite of admiration to the universal applause which your eminent qualities, as a philosopher and a politician, are so well entitled to.
I have the honor to be, with great respect
                                              Yours, etc,.
                                                                      The Count de Bruhl.

In the series, The Café de la Régence, by a Chess-player (published in Fraser's Magazine, 1840), it's noted that "Voltaire, the two Rousseaus, the profligate Duc de Richelieu, Marshal Saxe, Chamfort, St. Foix, Benjamin Franklin, Marmontel, Philidor, and Grimm, are but a few of the men of note who constantly frequented the Régence in early times."
While there's no mention anywhere of Franklin playing any notable figures at the Café de la Régence, it would seem likely that he did.

An anecdotal story concerning Franklin in the Régence has been  offered by Simpsons Contemporary Quotations, compiled by James B. Simpson, 1988:

Dr. Franklin was US Ambassador to France, and the center of the chess world was located at the Café de la Régence. None other than reputed world champion François Andre Philidor was a regular at the Café, and Franklin visited the Café in 1781 with the intention of having Philidor autograph his copy of one of Philidor's books on chess. Of course many visitors to the Café were making the same request to which Café proprietor Jacques Labar had a prepared denial to keep Philidor from constant interruptions. However upon recognizing the distinguished Franklin,  Labar promptly presented him to Philidor, who graciously autographed Franklin's book.
Once gone, Labar turned to Philidor saying, "François, you just autographed your book for the American Ambassador!" Philidor looked up from his game for the first time and said, "That's funny, I never knew that he was a chess player".

On the home front, Thomas Jefferson mentioned Franklin in a couple letters within a chess context

        in a letter to to Robert Welsh, December 4, 1818

"When Dr. Franklin went to France on his revolutionary mission, his eminence as a philosopher, his venerable appearance, and the cause on which he was sent, rendered him extremely popular. For all ranks and conditions of men there, entered warmly into the American interest. He was therefore feasted and invited to all the court parties. At these he sometimes met the old Duchess of Bourbon, who being a chess player of about his force, they very generally played together. Happening once to put her king into prise, the Doctor took it. 'Ah,' says she, 'we do not take kings so.' 'We do in America,' says the Doctor.
"At one of these parties, the emperor Joseph II, then at Paris, incog. under the title of Count Falkenstein, was overlooking the game, in silence, while the company was engaged in animated conversations on the American question. 'How happens it M. le Compte,' said the Duchess, 'that while we all feel so much interest in the cause of the Americans, you say nothing for them?' 'I am a king by trade,' said he."

                   and as quoted by John French in Travels in the USA and Canada, 1833

"On returning to the drawing room, we had a conversation which continued three hours, and the following were some of the sentiments Mr. Jefferson expressed. . . . I played with Dr. Franklin at chess, and was equal to him at the game."

Dr. Franklin's final and possibly greatest contribution to Chess was the publication of his essay on the Morales of Chess. It was published in Twiss' anecdotal collection in 1787, but Twiss credited Mr. H. Croft, the author of the Life of Dr. Young, for use of the essay:    The Morales of Chess