The earliest published text of Hamlet is dated 1603. It is a revenge tragedy, with which Shakespeare's audience would have been very familiar. It is worth researching this aspect a little further and asking yourself how closely Shakespeare follows the traditions of revenge tragedy and to what extent he develops and challenges the conventions of the genre.
Think about what was happening in English society around the time the play was written: the death of a long reigning monarch, challenges to established ways of viewing the universe and man's place in it, the discovery of the New World, political and religious unrest in Europe, etc. All in all this was a time of change and upheaval. Ask yourself whether Hamlet's dilemmas and continual questioning are a reflection of all this - an attempt to find answers to the big questions.
Hamlet is, perhaps, Shakespeare's most famous tragedy, and Hamlet himself his most famous tragic hero. Audiences over the years have differed in their responses to him. Do you see him as an indecisive and weak man who would never have been able to rule Denmark? Is he a noble and intelligent scholar who is destroyed by corruption and deceit? Is he a victim of Fate? Or is he too complex for the circumstances he finds himself in? What exactly would you say is Hamlet's tragic flaw?