The simplest answer is that we know practically nothing about his personality. If you were to type out all the known, verifiable facts about William Shakespeare, the text would fill less than half of one page:

He was born in 1564 (probably on April 23rd) and died in 1616 (definitely on April 23rd). He was born in Stratford-Upon-Avon to a family of modest but not insignificant means. His parents saw to his education as a child and he married Anne Hathaway (about whom we also know practically nothing) when he was 18 years old. She was 26 and pregnant with his first child (Susanna) on their wedding day in 1582. The twins, Hamnet and Judith, were born to the couple in 1585.

At some point in the 1580s he moved to London to be an actor before trying his hand at playwriting. His first plays appear on stage in 1589-91. By 1591 his work had become popular enough to attract the admiration and jealousy of his contemporaries (playwright Robert Greene called him an "upstart crow"). By 1596 he had produced at least two genuine hits (Richard III and Romeo and Juliet) that would be remounted numerous times throughout his career and he had purchased a 12.5% interest in The Lord Chamberlain's Men, serving as both actor and in-house playwright. In 1599 the Globe Theatre was completed and would serve as his primary artistic home for the remainder of his career. In 1603 the Lord Chamberlain's Men became The King's Men when King James I ascended the throne and became the group's primary patron. Shakespeare retired in 1613 or 1614 and moved back to Stratford where he occasionally collaborated with other playwrights until his death in 1616.

His will left most of his wealth to his daughter, Susanna. We have a few verified samples of his handwriting (mostly signatures) and he is mentioned in a few diaries and royal records for having performed in the private homes of wealthy families.

That's more or less it. We have no private correspondence, no personal diaries, and any mentions of him by his contemporaries either praise or deride his abilities as a playwright and actor, but never elaborate on his personality.

Everything that follows this sentence is in the realm of academic and literary speculation, extrapolation, and educated guesses.

A Midsummer Night's Dream is the only plot in Shakespeare that has no other literary precedent. The play's depiction of young people having romantic rendezvous in the woods at night is sometimes suggested as being an autobiographical depiction of Shakespeare's own courtship with Anne Hathaway in the wooded places around their home town. If that is indeed the case and the emotions of Midsummer's characters mirror his own, Shakespeare was quick to fall in love and when he did he fell hard. This assumption (along with information cherry-picked from the Sonnets) formed the basis for Tom Stoppard and Marc Norman's screenplay for the film Shakespeare In Love.

Shakespeare's plays express a wide (to put it mildly) variety of human opinion and emotion. His characters are sometimes described (particularly by Professor Harold Bloom of Yale University) as "invented humans" with all the subtle complexity and variation of real human beings. These same remarkable traits that make his plays unique also make them fruitless for separating the author's opinions from those of the characters. For example: I sometimes hear the assertion that all we can determine from the plays is that he didn't like dogs because nowhere in the cannon are dogs mentioned positively. This of course teaches us nothing, but it illustrates just how difficult it is to determine the individual traits of a man with such a gift for peering into the souls of others with nothing to go on but theatrical literature. All the plays really teach us about the man is that he had a staggering command of the English language (even inventing new words when his own were insufficient), a remarkable sense of human empathy, a terrific feel for comic and dramatic timing, and a chilling gift for philosophical introspection.

It is widely presumed that Shakespeare's sonnets represent his own intimate feelings and may not have been intended for publication. If it is possible to know anything about his personality and thought process, that's where we're most likely to find it. Unfortunately they offer more questions than answers.

The first 126 sonnets are addressed to an unidentified young male (the "fair youth"). The first 17 implore the young man to find a suitable wife and marry so that his virtues might live on after him. Then, in sonnet 18 ("Shall I compare thee to a summer's day?") the sonnets begin to take on a romantic tone towards the reader and there are occasional mentions of jealousy over a rival poet in #78-86. Sonnet # 20 explicitly laments that the fair youth is not a woman. These early sonnets, taken collectively, have sparked an argument among artists and historians as to whether sonnets 18-126 are evidence of a homosexual affair or if the love being expressed is entirely platonic.

Suddenly, beginning in sonnet 127 the poems are being addressed to a new subject: The Dark Lady. Sonnets 127-152 are unambiguously sexual in nature. #151 is particularly risqué and bawdy and is often cited to distinguish the relationships between the Fair Youth and Dark Lady to counter claims of Shakespeare's possible bisexuality. This portion of the sonnets is frequently used to suggest that Shakespeare was unfaithful in his marriage while in London at least once. His passion for his Dark Lady is the kind of smoldering, half-guilty hunger that seems more suited to a secret lover than a distant spouse.

I could go on and on about this topic and books about this very question are an industry unto themselves. Our lack of knowledge about Shakespeare's personality tempts us to project our own personalities and the conventions of our times on him, but the only honest answer is that we don't know and are not likely to ever find out.

For further reading:

Shakespeare: The Evidence: Unlocking the Mysteries of the Man and His Work by By Ian Wilson, 1993

Will in the World: How Shakespeare Became Shakespeare, by Stephen Greenblatt, 2004

Who's Who in Shakespeare's England by Alan and Veronica Palmer, 1981

Ben Jonson said of Shakespeare that "He was, indeed, honest, and of an open and free nature; had an excellent fancy, brave notions, and gentle expressions, wherein he flowed with that facility that sometime it was necessary he should be stopped. 'Sufflaminandus erat,' as Augustus said of Haterius. His wit was in his own power; would the rule of it had been so too. Many times he fell into those things, could not escape laughter, as when he said in the person of Caesar, one speaking to him: 'Caesar, thou dost me wrong.' He replied: 'Caesar did never wrong but with just cause;' and such like, which were ridiculous. But he redeemed his vices with his virtues. There was ever more in him to be praised than to be pardoned."

So we know that he was honest, not secretive, had a good wit, ran off at the mouth a bit when he had a clever conceit (Mercutio/Queen Mab), wasn't an entirely logical thinker (though we have to give him the ability to closely observe and record human behavior and follies, based on his literary output), but was generally a good person. Aubrey's comments (already given in this thread) indicate that he was not given to dissipation. Another comment by John Webster in his dedication to The White Devil (1612) portrays Shakespeare as being industrious.

We also know from history that he did not abandon his wife and family, and preferred Stratford to London, since he married there, bought a house, and raised a family there while working in London. From a letter written to him asking for a loan to pay expenses we know that he was willing to help his fellow townsmen. We also know that he was a good businessman and made investments with his earnings for his and his family's future security. In fact, as an actor, poet, playwright and theatre/acting company shareholder, we know that he gained most of his income from his theatrical investments, not from writing. We can also surmise with a good deal of confidence that he was concerned with the social status of his family, since his father's coat of arms was finally granted after he achieved some degree of financial success (and, as later events made clear, probably payment of a bribe, so we can gather that Shakespeare was a man of practicalities when the situation called for it), and his will indicates his desire to found a family dynasty.

Everyone loves to talk about how we have no understanding at all of Shakespeare in his day, but that's not entirely true. For example, here is a story from the diary of John Manningham, a contemporary of Shakespeare's in London:

Upon a time when Burbage played Richard the Third there was a citizen grew so far in liking with him, that before she went from the play she appointed him to come that night unto her by the name of Richard the Third. Shakespeare, overhearing their conclusion, went before, was entertained and at his game ere Burbage came. Then, message being brought that Richard the Third was at the door, Shakespeare caused return to be made that William the Conqueror was before Richard the Third.

Obviously it's a rumor, but it's a rumor that actually existed during Shakespeare's lifetime.

Just from watching the plays - he feels like a totally detached person. Amoral and atheistic (God does not feature at all in Shakespeare's world). Tickled by everything - Shakespeare is able to convincingly articulate the validity of a million contradictory viewpoints in one sweep - his greatness.