1) Hamlet bores for Denmark on the unheard-of notion that people die at the end of their lives, and that no matter how powerful or jolly they are, they'll eventually be reduced to skulls and dust. Horatio humours him with many 'You don't say?'s, but is looking at his watch the while. Hamlet is in danger of making Polonius look like Dorothy Parker here - if he hadn't already done so with his embarrassing 'advice to the players', so redolent of Prince Charles lecturing architects on the art of building design.
2) In passing, he mentions that he's engineered the murders of Rosencrantz and Guildenstern ("no shriving time allowed"), whose major fault seems to be that they obeyed the king's summons to come to Elsinore, and his request to try and find out what was upsetting Hamlet. Capital crimes indeed.
3) Within two seconds of realizing that Ophelia is dead - primarily as a result of his own actions - Hamlet is shouting at her grieving brother and fighting him. His reason? He dislikes Laertes' extravagant diction! (Or, more accurately, the fact that for once Hamlet isn't the centre of attention.)
4) Within a couple more seconds he says (the first and only time he expresses such a sentiment), 'I loved Ophelia'. After that it's all about him and how much more he can grieve than Laertes. '40,000 brothers' love' wouldn't equal his, insists this enemy of hyperbole. (At no point in the play is Ophelia alluded to again by either of them - even when they are forgiving each other for various other misdeeds.)
5) He takes time out to practise his favourite hobby of humiliating people who aren't in a position to answer back. Osric is an acceptable substitute for Polonius in this regard.
6) He denies his responsibility for killing Polonius by telling Laertes that he was mad at the time. (That, we remember, was the scene in which he told his mother: "My pulse, as yours, doth temperately keep time, / And makes as healthful music: it is not madness / That I have utter’d: bring me to the test, / And I the matter will re-word; which madness / Would gambol from.") Hmm.
7) Finally, when he is already mortally wounded, he does what he should have done at the end of Act I and kills Claudius. But even then he forgets his Princess Bride so far as to omit any mention of the fact that Claudius killed his father, concentrating instead on his supposed incest with Gertrude (at least in Q2, our copytext: in F he calls him 'murderous' but doesn't specify his victim). The whole thing is a mess, with bodies good and bad falling everywhere, and any sense of satisfied justice being wholly dissipated.
8) At the end of the play he argues for Fortinbras to become King. This passing of Denmark to a foreign power was what his own father had fought old Norway to prevent, and what the Danish army was guarding against at the start of the play - but Hamlet gives his country away, just because he likes Fortinbras's abs.
In sum, he is a self-centred, entitled, manipulative, untrustworthy, prevaricating, callous, incompetent little shit.
Now, bring on Paradise Lost!