I would ask how much he loved Ophelia. Was he willing to marry a girl with no royal blood line? Polonius is a courtier, but even Laertes warns her that Hamlet can't marry whomever he wants. He is a Prince after all.
One thing that never seems to get discussed, among all of those things we don’t know, about Hamlet is the relationship he had with King Hamlet. I would ask him about that. King Hamlet was a Catholic, old school fighting king. Hamlet was a scholar at the Lutheran school at Wittenberg. Did this cause conflict? Is this why Hamlet had not been elected by his father as the successor?
I would also ask if he might have had clues about the affair before his father died.
Which film version of your story do you like the best?
What did you like best at school in Wittenburg? What would you be if you weren't in line for the throne?
Were Rosencratz and Guildenstern equally guilty in your eyes?
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I would ask him several things:
Do you feel the problems in the royal family of Denmark were solved at the end of the play?
Do you think Fortinbras will be a good leader?
How much blame do you really put upon your mother?
Do you feel any remorse for your actions?
Do you feel any remorse for causing Ophelia's death, even if it was indirectly?
Why did it take you so long to kill Claudius? I know if I found out my uncle killed my father, the rage would over take me. How did you restrain it for so long?
Were there other way, besides the play, that you had thought of to convince yourself of Claudius' guilt?