In short: very little indeed. We don't even know the date of his birth, given that, according to the manners of the time, only the date of his baptism is recorded (2 February, 1585). William Shakespeare and his wife Anne Hathaway had three children, the eldest of whom was Susanna. Hamnet and Judith then followed—they were fraternal twins. It is probable that Shakespeare spent most of Hamnet's life away from Stratford upon Avon, where the family was based, because he was already very successful in London at this point. Hamnet died when he was eleven, in 1596. We don't know what specifically he died of, although some have suggested it was bubonic plague, drawing from the context of other death records in Stratford at the time.

In terms of what we actually know about Hamnet, then, that is about all, but this hasn't stopped scholars speculating for years about how Shakespeare was affected by his son's death and whether this is reflected in his plays. Obviously, having lost a child would allow Shakespeare to feel more keenly the grief of his characters in similar situations, but Shakespeare certainly didn't go on to write more tragedies after Hamnet died—if anything, he wrote a spate of comedies immediately after. There has also been a lot of speculation that Hamlet was written for Hamnett, but again this is without foundation beyond the similarity in their names.