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.