You could focus on the many film versions of his plays.  Ask if he's gotten a chance to see any of these (I'm assuming that if you can interview him, then maybe he could have the chance to catch a flick or two), then ask him what he thinks of the differences between film and stage.

You might consider asking him how he overcomes writer's block and/or did he really commiserate with women and their plight in his time (think of all the strong female characters and the ones who dressed as men in order to really live). 

Have fun with this...how much fun!

What an interesting assignment. You've already gotten a lot of good suggestions. Why not ask him what sort of play he plans to write next. Who are his influences? Which actor would he love to have in one of his plays? Since he is such an innovator in other ways, why not be really innovative and use female actors?

Many author's first work seems to be autobiographical.  Given that, I'd like to ask him which character he created is MOST like himself. Perhaps all the plays are based on "external" sources, but maybe there is one characters (especially in the tragedies) that is more a part of "him" than any of the others

Shakespeare's perspicacity is amazing in all his plays.  I would inquire how he attained such insight into the human psyche and heart at his relatively young age.

"O what fools these mortals be!" exclaims Puck in Twelfth Night as he reflects upon the foolish errors of jealously and infatuation.   Then, in Hamlet Shakespeare proves the foolish error of the deadly sins of lust, pride, and anger. 

Perhaps in answering the question about his understanding of human nature, Shakespeare would then give the interviewer a "sneak preview" into the themes of Hamlet.

One of the most interesting things about Shakespeare is the more or less total absence of biographical information we have about him.

Questions about his authorship, about his family (for the most part, at least), about his specific likes and dislikes, or even (according to the Sonnets, some critics would argue) his sexual preference are ones you simply couldn't provide the correct answers to: it's going to be an imaginative exercise. Though, for what it's worth, I'd love to know why he only left his wife, Anne Hathaway, his second-best bed in his will? Remember too that Shakespeare buried his son Hamnet (name notably similar to Hamlet's!) in 1596, which might provide some interesting questions.

Two interesting questions you could tentatively answer with scholarly opinion on your side are:

• Did you act in your own plays? (Shakespeare is listed in the First Folio as one of the actors, so the answer is a definite yes - scholars think he played Polonius in "Hamlet", and the title role in "Julius Caesar".
  
• What costumes were worn in productions of your plays? (Look at the Peachum drawing of "Titus Andronicus", which shows actors wearing Elizabethan costume - the modern dress of the day - with Roman 'accessories').
  
  

Another aspect of Shakespeare you could focus on would be the question of the authorship of his plays.  You could ask him why he thinks that people don't think he could have written the plays, and then you could ask him to respond to those questions.  

For example, there are some that question that he didn't have the proper education to read/understand the languages needed (Latin, Italian, etc.) to have used some of the sources he based several plays on (the Roman history plays, "Romeo and Juliet", etc.).  

You could also ask his opinion of some of his contemporaries that historians have suggested really did write the plays (The Earl of Oxford, Francis Bacon, etc.).

It shouldn't be too difficult to find sources for all of this.

It is 2008 so why not go for the personal stuff? You could ask about his wife and children. Are they supportive? What's his relationship like since they live so far away?

Other options could be the political landscape. How helpful has patronage been to his writing? How does he respond to critics?

Who are his main influences? What contemporaries (other playwrights or writers) does he admire?

The answers to these types of questions are readily available with a bit of research.