Claudius begins his soliloquy by portraying his "offense"—killing his brother, Old Hamlet—as "rank," i.e., foul-smelling and repulsive. Claudius want to pray, but his guilt is so overwhelming that he is hesitant to even contact God, considering the gravity of his crime. So, he talks to himself in order to address his fear of punishment.
In the first part of his speech, Claudius tries to justify his action by saying that he did it out of mercy for his father. He also mentions that Prince Hamlet was not responsible for his mother's death, so killing him would be unjust. Finally, he claims that he killed King Hamlet in order to protect Denmark from his madness.
However, all these arguments are weak because they are based on what Claudius calls "false premises." He says this when explaining why he thinks killing King Hamlet was wrong: "For to kill a king, 'tis not enough / That we do take his life; but we must cause / His death to have merit of justice" (I.4.78-80). In other words, killing a king should be a very serious offense because it violates one of royal rule's most important principles: that rulers are above murder.
This means that Claudius's claim that he killed King Hamlet in order to protect Denmark is false, since protecting someone by murdering them would make their death meaningless.
For the first time in Hamlet, Claudius' guilt is definitely confirmed by the phrase "O, my crime is rotten, it reeks to heaven." Claudius begins his soliloquy by portraying his "crime," the murder of his brother, Old Hamlet, as "rank," i.e., foul-smelling and disagreeable. This image contrasts sharply with Claudius's earlier description of himself as a "sweet and gracious king" who deserves to be praised for his virtues rather than punished for his sins.
Claudius's remorse over the death of Old Hamlet is evident from his use of the word "offense." This word has two meanings when used by Claudius: it can mean either "sin" or "crime." In this case, it means "sin" because Claudius believes that the only reason why he was convicted by his brother's ghost is because he committed a serious sin by murdering him.
Claudius takes pride in thinking that his crime is not very grave because it does not cause much harm to others. However, like most people, he fails to realize that something is wrong even if no one is harmed physically. Religion teaches us that our sins can never be wiped out even though they may not always have visible effects. God's justice is not based on how much damage we cause but on our intentions.
In Act III, Scene III, Hamlet appears to be ready to act on his desire for vengeance. He is pleased that the play has shown his uncle's culpability. The audience is given actual confidence that Claudius murdered his brother when he prays: a full, spontaneous confession, even though no one else hears it.
Claudius dies with this prayer on his lips. It is safe to assume that he confessed before he died.
This scene was probably written after Claudius' death. Otherwise, Hamlet would not have known about it. However, earlier in the play we are told that Gertrude and Claudius did not know each other before they married. So, how could she have heard him confess? Did someone tell her after she married him? No, since she had just lost her husband, it makes more sense that she ignored or forgot about this incident. She may have been upset that night but then got over it. Also, there is no evidence that shows that Claudius killed himself out of guilt. He most likely did so because he knew he was going to be accused of murder and wanted to spare Gertrude the pain of hearing it from others. He may have also done it to escape the madness of his father's curse.
Finally, this scene was most likely not in the original version of the play. There are many changes between the original version and the one we see today.