I will just pick the difference between two soliloquies, or pieces of them to suggest the character changes he has gone through between them. The first is in Act I, scene ii and the second is at the end of Act II, and they show some pretty clear changes in Hamlet's character.
The opening lines of the first soliloquy paint a picture of Hamlet as a completely static person, unable to make any sort of decision and mainly contemplating suicide as a way out of any responsibility for action:
O, that this too too sullied flesh would melt,
Thaw and resolve itself into a dew,
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter! O God! God!(135)
How weary, stale, flat and unprofitable
Seem to me all the uses of this world!
The soliloquy at the end of Act II gives a completely different view of our prince, one who has seen now more of the causes and underlying treachery present in Denmark and ahs decided to take action, to find proof of his uncles treason:
For murder, though it have no tongue, will speak
With most miraculous organ, I'll have these players
Play something like the murder of my father(590)
Before mine uncle. I'll observe his looks;
I'll tent him to the quick. If he but blench,
I know my course. The spirit that I have seen
May be a devil; and the devil hath power
T' assume a pleasing shape; yea, and perhaps(595)
Out of my weakness and my melancholy,
As he is very potent with such spirits,
Abuses me to damn me. I'll have grounds
More relative than this. The play's the thing
Wherein I'll catch the conscience of the King.
For though the murder cannot actually tell its story itself, Hamlet has finally decided to do something about it. He will get proof, as best he can, by using the play and examining the reaction of the king.
So now he has become a man of action rather than inaction, he has some resolve instead of just whining (according to some) about the terrible nature of everything and the fact that he can't even commit suicide to get out if it.