In Hamlet, after worrying about the problem of action for 4.5 acts, Hamlet finally takes vigorous action. What are the motives for these acts? How do they relate to his initial charge of revenge? For example, why does he kill Claudius?
Hamlet and Claudius have been engaged in a duel of wits from the beginning. Claudius is afraid Hamlet is plotting a coup and is trying to pry into his stepson's mind to find out what he is thinking. Hamlet isn't planning anything until he talks with his father's ghost and is told that Claudius murdered his father and usurped the throne. Hamlet seems reluctant to act against Claudius because he isn't sure the Ghost wasn't the Devil in disguise, but he decides to act insane because he is afraid that Claudius will see how much he has changed and will become even more suspicious and inquisitive. A number of men also saw the Ghost. Evidently Hamlet has sworn them all to secrecy--but if Claudius started questioning them and comparing their stories, and even threatening them with torture or death, he might find out that the ghost of the man he murdered has been visiting the castle and has been talking to his son Hamlet.
There are no deaths in the play until Hamlet kills Polonius in his Gertrude's bedroom in Act 3, Scene 4. This precipitates a crisis. Claudius decides to do what he has been thinking of doing all along: he sends Hamlet to England with Rosencrantz and Guildenstern bearing a sealed letter demanding that the English decapitate Hamlet as soon as they read it. Hamlet is intuitive. He reads the letter and substitutes another calling for the English to decapitate Rosencrantz and Guildenstern instead. These two toadies go off to England, but Hamlet is accidently seized by pirates, ransomed, and ends up back in Denmark. Now he realizes that the battle of wits with Claudius has become a life-and-death struggle. If he doesn't kill Claudius, Claudius is going to kill him. And Claudius is an absolute monarch who can have anybody killed just by ordering it done.
Much of this is learned by the viewer through what Hamlet tells his friend Horatio, who says:
It must be shortly known to him from England
What is the issue of the business there.
In fact, an Ambassador from England arrives that very day to inform Claudius that his orders to behead Rosencrantz and Guildenstern have been carried out.
Hamlet replies to Horatio:
It will be short: the interim is mine;
And a man's life's no more than to say 'One.'
But I am very sorry, good Horatio,
That to Laertes I forgot myself;
For, by the image of my cause, I see
The portraiture of his: I'll court his favours.
But, sure, the bravery of his grief did put me
Into a towering passion.
It is ironic that Hamlet speaks so favorably about Laertes when Laertes and Claudius have conspired to kill him and Laertes intends to do so with a poisoned fencing foil.
Hamlet is taking "vigorous action" because his life is in danger. He knows the King wants him dead but does not know exactly how the King intends to bring it about. But when he says, "It will be short: the interim is mine," he means that he can and will act first. Claudius, however, is one step ahead of him. He invites Hamlet to a fencing match with Laertes and has even prepared a poisoned drink to offer his stepson in case Laertes' poisoned foil fails to do its work.
All the principals in the play end up dead, Hamlet last of all. Polonius is first. Rosencrantz and Guildenstern are beheaded in England. Ophelia drowns herself accidentally or on purpose. The others all die in the last act. Laertes stabs Hamlet and is stabbed with the same foil. Gertrude drinks the poisoned cup intended for Hamlet. Hamlet stabs Claudius and then dies from Laertes' poisoned foil.