I would ask Shakespeare if Hamlet's dad really did appear to him as a ghost, or if Hamlet was mentally ill. I would also ask him, if I am permitted another question, why Romeo and Juliet had to die. They were star-crossed lovers, but they could have lived happily ever after. They might have just woken up, and their parents still would have gotten the message.
I would want to ask about his attitude towards fate and religion. In so many of his plays characters seem to be subject to the whims of fate and do not act to try and solve their situation or find a way out. I would like to know if this was Shakespeare's thinking about fate and whether he saw himself as an "actor" against this backdrop or as a passive victim.
I agree with lynn30k -- I'd want to know how much of each play he wrote, and what percent he "borrowed" or lifted from other plays -- which was common and accepted practice at the time. I'd also like to know where he got such a great depth of education, and how he knew the human heart so well.
Did you really write all the plays attributed to you? There has been some debate on this topic.
I would ask Shakespeare; What was your favorite play? I have always thought that "The Tempest" was Shakespeare's favorite play, but really have no basis for it. I would like to know which of his plays Shakespeare thought was the most valuable.