First of all, there is the ever-important death of his father, and his mother's marriage to his uncle only weeks afterwards. That event sends Hamlet into a downward spiral of rage and depression, causing him to mope about the castle, break up with his girlfriend, reject his friends, and want to kill himself, essentially. Those events are the foundational groundstone for all else in the play; they help to define him as a deeply emotional and disturbed young man.
Then, we have the fact that he sees his father's ghost and is told by said ghost that he was murdered by the hated uncle and to top it all of, he wants Hamlet to enact revenge. That shakes Hamlet's world, first of all by revealing that his uncle is a conniving murderer and manipulator of women, and secondly, by giving him command to commit murder himself. He spends the rest of the play mulling over this task of revenge, and trying to decide how, when or even if to do it. The entire play is centered around Hamlet's moral quandary and hesitation to act out that revenge; this incident defines him as a contemplative, hesitant and thoughtful person who is tentative about killing someone.
The next significant event is the play that he stages, in which it becomes quite clear that his uncle is guilty of murder. This event defines Hamlet in a couple ways. First of all, it defines him as clever; he was pretty smart to set up that play. Secondly, it shows that he wants to be totally sure that the man is guilty before taking his life. This defines Hamlet as a moral, conscience-driven person who has set boundaries and rules of conduct.
Another event that defines Hamlet is his refusal to kill Claudius while he is praying. This shows that Hamlet is a very religious person who has the fear of God in him, and that he truly believes in Hell, and wants to ensure Claudius goes there. It also defines him in a negative way as it highlights his cowardice; he is assured of his uncle's guilt and has the perfect opportunity to kill him, but doesn't. He's kind-of a chicken.
I think one significant even that often gets overlooked is his reaction to seeing the dead Ophelia being lowered into her grave. In a moment of true candor, he jumps into the grave and declares that he loved her with all his heart, more than any brother could have loved her. This is incredibly defining, because it shows that when he was rude to Ophelia before, it was all a show, a farce, and that he wasn't being sincere. It also shows him revealing a bit of sadness and remorse over the fact that his actions might have harmed someone else. We see him exposed, vulnerable, and for once, not putting on a show for someone else's benefit. It defines him as a deeply feeling individual.
I hope that those give you a couple ideas; good luck!