Hamlet lives in an almost perpetual state of disillusionment. His father left him, albeit through no fault of his own, and he understands death can strike unexpectedly. His mother remarries-- his uncle!--in an extraordinarily short time AND acts as if she cares for him. Hamlet is disillusioned about love as well as his mother. He discovers Claudius killed his own brother for the throne and the queen, and Hamlet has to question love of any kind. His two childhood friends become spies for his murderous uncle, turning them into traitors toward his friendship. And the list goes on.... It is true that Hamlet already probably had a rather melancholy bent when we first meet him, but these consistent disillusionments no doubt add to his depressed state of mind.
I would think that you would really want to emphasize his disillusionment with his mother. That seems to really eat at him, as seen when he rants at her in Act III, Scene 4.
You might also talk about Hamlet's disillusionment with himself. He, in my opinion, becomes more and more disenchanted with his own inability to act.
So if your mother can't be trusted and you can't be trusted yourself, that's pretty strongly disillusioning, I think.
To start, I'd say that the entire play is about struggle. Hamlet struggles with what has happened to his father, the fact that his mother married his uncle and with his love for Ophelia. All of that is struggle and "man vs. man" conflict. The important point of your discussion, in my opinion, would be the disillusionment.
By definition, disillusionment is "to cause to lose naive faith and trust" according to Merriam-Webster. Hamlet is disillusioned with his royal family once he finds out from his father's ghost that he was murdered by his uncle. He's also disillusioned with his parents' marriage, because of the fact that his mother went to her brother-in-law so quickly. You can also bring in Rosencrantz and Guildenstern in that he catches them lying to him about why they are at Elsinore. No longer is his friendship with them real. You can also bring in his relationship with Ophelia, because she decides to give him his "rememberances" back, ending their relationship. I think he realizes that she's going along with what her father says and therefore, he is disillusioned by what he thought was their relationship.