So, a quick recap. Hamlet has just put on a play, that he calls “The Mouse Trap,” to try to see Claudius’ reaction to the way he killed his brother, King Hamlet. Hamlet then asks his friend, Horatio, to stand watch to try to catch Claudius’ reaction or any other clues or expressions he might show. When the murder goes down on stage, Claudius rushes out of the room, which confirms Hamlets suspicions to himself: Claudius is guilty of King Hamlet’s murder.
One reference that is interesting, is this one in Hamlet’s soliloquy at the end of the second scene:
O heart, lose not thy nature; let not ever
The soul of Nero enter this firm bosom.
Let me be cruel, not unnatural;
I will speak daggers to her, but use none.
Nero is a roman emporer who, among other things, is infamous for killing his mother. Here, when Hamlet hopes that the soul of Nero will never enter his firm bosom, he means that he hopes he will never become so overcome with rage that he will hurt his mother in any way. This also refers back to Hamlet’s first meeting with the ghost of his father, who specifically said that no harm is to come to the queen. He does say that he will “speak daggers to her,” meaning that he will try to guilt trip her, which he does pretty effectively in the beginning of his play. His mothers guilt prompts her to make a comment and even summon him at the end of the scene.
The ending of the third scene is a little unfortunate for Hamlet also. He goes to kill Claudius, but seeing him praying stops. This is because he does not want Claudius to have the justification of prayer or the forgiveness of God when he dies. So Hamlet leaves. Right after this though, Claudius comments that his prayers were useless because he didn’t mean them, and feels no true guilt or repentance. Basically, Hamlet could’ve killed him right then and there but misunderstood the situation and didn’t.