What action does Claudius take against Hamlet after the play in act 3?

After Hamlet stages the "mousetrap" play, Claudius knows beyond the shadow of a doubt that Hamlet is aware that he, Claudius, killed the senior Hamlet to get the throne. Claudius now knows the reason why Hamlet has been acting so strangely. Then, after Hamlet kill Polonius, Claudius has to assume that he was the target, not Polonius. He also recognizes that Hamlet is capable of killing. Claudius understands he has to act fast and get rid of Hamlet for good.
Hamlet is popular in Denmark, so rather than jail him for murdering Polonius, Claudius sends him to England with his courtiers Rosencrantz and Guildenstern. He sends the duo with a sealed letter ordering that Hamlet be killed when he gets to England.
Since the two courtiers are fools, it doesn't take long for Hamlet to open the letter, read it, and write a new letter condemning Rosencrantz and Guildenstern to death. Having dodged this bullet, Hamlet returns to Denmark. So does Laertes, bent on killing Hamlet to avenge his father. Claudius manipulates the hot-headed Laertes into fighting Hamlet with a poisoned rapier.
Claudius does not last long as a monarch. If Hamlet had not killed him at the end of the play, he would have been deposed by Fortinbras arriving with an army. As it happens, Hamlet is wounded with the poisoned rapier and dies, but it does Claudius no good.

In act three, Hamlet has a private conversation with Gertrude and kills Polonius, who was secretly listening in on their conversation. Following Polonius's death in act 3, Claudius believes that he will be held responsible for Polonius's death, and he moves forward with his plan to send Hamlet to England alongside his friends, Rosencrantz and Guildenstern. Claudius also secretly sends a letter with Rosencrantz and Guildenstern instructing the authorities in England to murder Hamlet upon his arrival. However, Hamlet ends up discovering Claudius's letter and returns back to Denmark. Claudius then plots Hamlet's death by setting up a fencing match between Hamlet and Laertes, who wishes to avenge his father's death. Claudius also has Hamlet's drink poisoned as a backup plan in the event that Laertes fails to strike the prince with his poison-tipped sword. Tragically, Claudius's plan backfires as Gertrude drinks the poison, and Hamlet kills Claudius before dying alongside Laertes.