In Act 4 Scene 3 of Hamlet, what are the details of the king's plan to execute Hamlet ?
By this point in the play, Claudius has revealed his guilt in killing King Hamlet, and Hamlet has revealed that he knows the truth, so now it is a matter of who is going to survive these revelations. Claudius' plan is to tell Hamlet and the court that he being sent to England to retrive some tribute that England owes to Denmark. The truth is that Claudius will have Hamlet deliver a letter "conjuring to that effect, the present death of Hamlet."
Throughout Act 4 Claudius refers to Hamlet as a disease that must be cured. In scene three he says "Diseases desparate grown / By desparate appliance are relieved, / Or not at all." He knows that having Hamlet killed is an extreme action, but he is desparate to maintain his throne and his wife, so he will have the threat of Hamlet eliminated by an external force. He knows he can't do anything himself against Hamlet because it would destroy Gertrude, so his final words, again using the disease imagery, are "Do it, England; For like the hectic in my blood he rages, And thou must cure me." Claudius hopes that this trip to England and Hamlet's death put an end to the situation.