There are a number of different ways you can approach this. My own personal recommendation would be to combine both Hamlet's loneliness and depression with a study of his various friendships, and the way that the betrayal of those friendships exacerbates Hamlet's sense of isolation. Let us consider how Hamlet is feeling. He is lost, alone and incredibly depressed about his father's death and his mother's marriage to his uncle, which has, incidentally, disinherited him from the position that should by all rights be his. However, what adds to this terrible situation is the way in which all of his closest friends and confidantes, bar Horatio, one by one betray him and show their closer allegiance to the King. Consider Rosencrantz and Guildenstern in Act II scene 2. Again and again, Hamlet asks them why they have come to Elsinore, only to be given evasive answers. In the end, Hamlet himself works out the truth:
But to the purpose; you were sent for; and there is a kind of confession in your looks; which your modesties have not craft enough to colour, I know the good King and Queen have sent for you.
If this betrayal of his two old friends was not enough, in his famous confrontation with Act III scene 1, some productions show that Hamlet realises that Ophelia has been set up by Polonius and Claudius against him. This is what leads to his virulent and bitter "Get thee to a nunnery!" speech. Again and again, Hamlet is shown to be betrayed by those closest to him, leaving him ever more alone and depressed. This would make an excellent basis for an essay on Hamlet's character.