I believe the most important lines in this soliloquy are the following:
I'll have grounds
More relative than this: the play 's the thing
Wherein I'll catch the conscience of the king.
It is clear that Hamlet has finally become determined enough to prove Claudius' culpability in his father's murder and intends using a play to show him up. Hamlet is convinced that he can use the play, in which he would depict a murder much like that of his father, to expose his uncle. He believes that Claudius will react to the events he intends to depict in the play, thus displaying his scheming and his reprehensible act. Hamlet says earlier in his monologue:
I have heard
That guilty creatures sitting at a play
Have by the very cunning of the scene
Been struck so to the soul that presently
They have proclaim'd their malefactions;
For murder, though it have no tongue, will speak
With most miraculous organ.
Once, he believes, he has the ultimate proof of his uncle's guilt, he would have no recourse but to kill him in revenge. Up to now, Hamlet has not been entirely certain about what to do. He has tarried and is, in this soliloquy, overwhelmed by his failure to act decisively. He contrasts his actions to those of an actor who, he believes, is more capable of evoking a reaction than he has been able to, up to now. Hamlet is profoundly critical of himself here, calling himself 'an ass' who has not made a move though he has had much reason to do so.
Hamlet doubts his courage, asking whether he is a coward who cannot even act against the abuse he suffers at the hands of Claudius, who is a:
bloody, bawdy villain!
Remorseless, treacherous, lecherous, kindless villain!
He feels that Claudius has taken advantage of his weakness (his indecision) and his sadness (for his father's demise) and has mistreated him. However, Hamlet now feels more confident and is desperate to finally end his strife and take his uncle's life - a dramatic turning-point in the play.