What characters display guilt in Shakespeare's Hamlet? What are the quotes that indicate it?
Just as dstuva said so well above, the most guilty individual in the play is King Claudius. Although he never comes right out and says he is sorry for what he has done, he cannot pray and knows full well that he is a guilty murderer and despicable liar.
And, yes, Hamlet, too, feels guilty for not acting with more haste to exact the revenge that his father's ghost so sorely needs.
But there is one more person who, although she can perhaps be excused for her weakness, also bears some guilt and feels it. That person is Hamlet's mother, Queen Gertrude.
After the play-within-the-play and just after Hamlet spares Claudius' life while he is kneeling in ineffectual prayer, Hamlet goes to see his mother in her chamber. He is pretty rough with her and rightly so; she has, out of weakness and ignorance, married her dead husband's killer. The ghost intrudes on the scene in order to prod Hamlet to get on with his revenge, and then Hamlet says to his mother (Act 3, Scene 4):
...It is not madness
That I have utt'red. Bring me to the test,
And I the matter will reword; which madness
Would gambol from. Mother, for love of grace,
Lay not that flattering unction to your soul
That not your trespass but my madness speaks.
It will but skin and film the ulcerous place,
Whiles rank corruption, mining all within,
Infects unseen. Confess yourself to heaven;
Repent what's past; avoid what is to come;
And do not spread the compost on the weeds
To make them ranker. Forgive me this my virtue;
For in the fatness of these pursy times
Virtue itself of vice must pardon beg
Yea, curb and woo for leave to do him good.
O Hamlet, thou hast cleft my heart in twain.
And with that last line, Gertrude, at least in part, admits to some guilt in her actions. She then comes to believe in her son and his cause and agrees to do her best to, finally, act with honor and intelligence.
In the final act, Hamlet feels guilty for having procrastinated regarding his avenging the death of his father, King Hamlet. In Act V, he says to Horatio,
Sir in my heart there was a kind of fighting
That would not let me sleep. Methought I lay
Worse than the mutines in the bilboes. Rashly,
And praised be rashness for it--let us know,
Our indiscretion sometime serves us well,
When our deep plots do pall;and that should learn us
There's a divinity that shapes our ends,
Rough-hew them how we will-- (V,ii,4-11)
It has been argued by critics that Gertrude, the Queen and mother of Hamlet feels guilt over her having married her husband's brother, Claudius, and married him so swiftly after the death of her husband. Certainly, the castigation of Hamlet has caused her to feel some shame, if not guilt. In the "closet scene" of Act III, she expresses remorse for her behavior near the end of this act:
Be thou assured, if words be made of breath
And breath of life, I have no life to breathe
What thou hast said to me. (III,iv,201-203)
In the final act, when Gertrude drinks from the cup of poisoned wine intended for Hamlet, she tells him it is poisoned as she dies, an act which can be interpreted as her wish to acknowledge the evil in Claudius and atone for her wrongdoing in marrying him and in thinking Hamlet mad. Indicating her misery and guilt, Hamlet tells her goodbye as she dies: "Wretched queen, adieu!" (V,ii,312)
Hamlet and Claudius are probably the only characters that feel at all guilty in Shakespeare's Hamlet, and Claudius's guilt is more like worry over his eternal salvation, or lack of it.
Hamlet repeatedly rebukes himself for his failure to enact quick revenge upon Claudius for his father's murder, as the Ghost tells him to. You can find quotes showing Hamlet's guilt in his soliloquy starting at line 515, in Act 2.2, in which he calls himself a coward and a villain. You can also look at his soliloquy in Act 4.4, starting at line 32, in which he talks about all occasions pointing toward his failure to do what he's supposed to do.
Claudius shows guilt, of a kind, after he sees the play within the play and realizes Hamlet knows the truth about his father's death. He is praying and talks about his offence smelling clear to heaven (Act. 3.3). Notice, though, that he never really confesses, and never really says anything about being sorry for what he's done. He acknowledges his guilt, but is not really sorry for it.