1.2 We meet Claudius during an official ceremony where he explains the slightly awkward situation that brought him to the throne. He then goes on to deal capably with the day-to-day business of the court.
1.2 Claudius tries to convince Hamlet to stop mourning for his father.
2.2 Diplomats announce that war with Norway is averted. Polonius trots out his "mad love" theory of Hamlet.
2.3 Rosencrantz and Guildenstern prove unhelpful, but overhearing Hamlet and Ophelia makes Claudius nervous. It doesn't sound like love is the thing making Hamlet angry – it's more like marriage is. "Those that are married (all but one) shall live," Hamlet says. Not a good sign for Claudius, or, in fact, newlyweds everywhere.
3.2 Claudius watches Hamlet's play, sees a reenactment of the way he killed his brother, freaks out, and runs from the room.
3.3 Claudius prays for forgiveness for killing his brother, but realizes he's not willing to give up the crown or the wife he got from his foul deed. In this way, he unknowingly escapes being killed by Hamlet.
4.1 Claudius finds his wife totally upset after her confrontation with her son. Polonius is also dead, which makes him not so happy. Claudius decides that Hamlet must be sent away to England now. This way they can start damage control.
4.3 Claudius gets Hamlet to tell him where Polonius' body is. Once the Prince is gone, the King reveals he has ordered for Hamlet to be killed once he gets to England.
4.5 Despite the castle being totally invaded by followers of angry Laertes, Claudius manages to calm everybody down and keep the crown on his head.
4.7 Claudius finds out that Hamlet is still alive. He plots with Laertes about how they're going to kill Hamlet, supposedly so Laertes can have his revenge. Gertrude enters with news of Ophelia's death, which gets Laertes all worked up again. This frustrates Claudius to no end.
5.1 Claudius attends Ophelia's funeral, where Hamlet shows up and makes trouble.
5.2 After Claudius fails to stop his wife from drinking a goblet of wine he knows is poisoned, he gets a taste of his own medicine – literally. Hamlet runs him through with a sword and then makes him drink the rest of the poisoned wine.