The first thing Claudius reveals in his soliloquy is the most significant: he admits that he has murdered his brother. Up until this point in the play, we have had only the Ghost's accusation of murder. Now we hear the revelation directly from Claudius himself:
O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder!
Claudius then reveals that he cannot pray, despite his attempts, and imagines his own hand (in an image echoing Macbeth) "thicker than itself with brother's blood". Claudius then reveals his belief that he cannot be forgiven because he still covets and keeps the things for which he committed murder; revealed explicitly, they are:
My crown, mine own ambition, and my queen.
Claudius believes he cannot be forgiven without giving up these things and, as he has no intention of giving them up, calls upon the angels to help him and forgive him.
Yet what can it when one can not repent?
O wretched state! O bosom black as death!
O limed soul, that, struggling to be free,
Art more engaged! Help, angels! Make assay!
Bow, stubborn knees; and, heart with strings of steel,
Be soft as sinews of the newborn babe!
In this soliloquy, Claudius confesses that he has murdered his brother, Hamlet's father. Up until this point, we have had the ghost's testimony that this is so, which as Hamlet understands, may be unreliable: the ghost could be Satan in disguise, tempting Hamlet to murder an innocent. Just prior to this scene, Hamlet staged the mousetrap play and though Claudius's reaction to it suggested guilt, this is first time we have full evidence that he committed the crime.
Why we love Shakespeare is revealed in this soliloquy: Claudius is an evil character who has done an evil deed that he doesn't repent of, but he also has enough of a moral compass to realize he has done an evil thing. He doesn't rationalize away what he has done. He doesn't say his brother deserved it or he had no choice but to kill him: he completely owns his guilt. He spends much of the soliloquy taking about how he can't really pray for forgiveness, because God, who sees into hearts, will know he is lying. If in the corrupt Danish court, he could buy injustice, this is not so in heaven:
But ’tis not so above.There is no shuffling. There the action liesIn his true nature, and we ourselves compelled,Even to the teeth and forehead of our faults,To give in evidence. What then? What rests?Try what repentance can. What can it not?Yet what can it when one can not repent?
This honesty about his own evil complicates and fleshes out Claudius as a character, which is what Shakespeare does so well. Claudius wishes he were a different person, a better person more acceptable to God, and is in agony over this, but accepts that he isn't, and even that, when push comes to shove, he wants to be the evil person he is more than the good person he knows he should be. He wants to want to be good, as he says earlier in the soliloquy:
Pray can I not.Though inclination be as sharp as will,My stronger guilt defeats my strong intent,And, like a man to double business bound,I stand in pause where I shall first begin,And both neglect.
Isn't this human nature in a nutshell? He wants to be good and pure, but he also wants the fruits, such as power and his brother's wife, that evil brings. At the end of the soliloquy, Claudius decides he will try to pray though he assumes it won't work; nevertheless from his conflicted soul he calls on supernatural power for help. This brings dramatic irony into the action: we as an audience know what Hamlet does not when Hamlet comes across Claudius praying. Hamlet refrains from killing him, figuring Claudius would be in a state of grace and go to heaven if killed at prayers, but we know that Claudius most likely is not able to repent and pray for forgiveness: at this moment, Hamlet could have killed his uncle and it's very possible his uncle would have gone to hell.