I don't think there is any way to know, either way. However I would argue that the society Shakespeare lived in was so different from our own that it would be difficult to describe or judge anyone from then, because we would be using modern eyes.
As with so many questions regarding Shakespeare, this one has no definitive answer and probably never will have. Some have argued that the so-called 'dark lady of the sonnets' was actually male and that that was the real mystery about her/him. And of course much can be read into the plays by anyone with a mind to. For example, Antonio in 'The Merchant of Venice' is very much the rich patron at the centre of a kind of boys' club and his very strong affection for Bassanio could be seen to be homoerotic. Other plays too, even 'Hamlet', can bear homoerotic interpretation, but of course this would still be very hypothetical and would tell us nothing definite about Shakespeare himself. There is also the fact that the theatre world of Shakespeare's time would have been attractive to homosexual males, even more so than today. It was an exclusively male habitat and, since females were excluded, the more effeminate males probably played the female roles. In such circumstances it is entirely possible that Shakespeare was homosexual or bisexual but conclusive proof either way will probably never emerge. And does it matter anyway?
While history indicates that Shakespeare was in fact married and that he had children, there remains speculation about his actual sexuality in literary circles. One must remember that at this point in history, it was not uncommon for men to engage in homosexual acts with other men, even though it was somewhat frowned upon by "proper" society.
There have been speculations that Shakespeare had, at one time, a male lover, but there is no "hard-and-fast" evidence proving this in the same way that his marriage and fatherhood have been. A good amount of the sexuality questions surrounding Shakespeare are merely a result of his own proficiency as an author and playwright. In this way, people tend to stereotype and generalize creative artists as "gay" or "sensitive." Shakespeare is no different.
Was Shakespeare really gay?
Were William Shakespeare's sexual preferences toward men or women?
All great creative artists are capable of complete identification with the point of view of any character they create or, indeed, pretty much anyone they care to think about. Shakespeare is especially remarkable (see Harold Bloom) for creating entirely self-conscious characters - there are no cardboard figures in mature Shakespeare. He could see them all from the inside. Therefore, though I believe everyone is bisexual with a bias of inclination formed by habit, society and opportunity, great creative artists are especially so. And thus the sexual orientation of his characters is always completely convincing, from the explicitly straight to the implicitly gay (of whom there are, naturally, very few: Achilles and Patroclus, the sea captain in Twelfth Night, Antonio in Merchant of Venice, etc.).