His first plan is to send Hamlet off to England along with Rosencrantz and Guildenstern. R & G will have orders to have the English execute Hamlet. That doesn't work out, however, so Claudius then proposes to make use of Laertes's desire to avenge his father's death, and have Laertes kill Hamlet. Or if that doesn't work out, Claudius proposes to poison Hamlet.
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The King proposes to rid of Hamlet by sending him to England. In England, he would then have Hamlet murdered.
Because by that point King Claudius knows that Hamlet knows the truth about king Hamlet's death, so he tries to get rid of Hamlet using Laertes.
Hamlet is directly responsible for the deaths of five people and indirectly for another. The first person to die because of Hamlet's actions is Polonius, who Hamlet stabs in the mistaken belief that he is King Claudius. This indirectly causes Ophelia's death. Hamlet then kills, in succession, Rosencrantz, Guildenstern, Laertes and Claudius.
He wants to get rid of him.
Well, I must first of all say that my statement is of my own interpretation of the story. However, it may be possible that Claudius was planning on killing off Hamlet (or at least spy on him) from the very beginning. By looking at Claudius's personality, namely his willingness to kill for power, it can be implied that Claudius is fearful of loosing his power in the same way. Of all that may pose a threat to Claudius, Hamlet seems to be the most likely threat as he is in direct line to the throne (after all why not kill off the king and become the new king a little ahead of schedule?). So due to Claudius's paranoid nature, it may be possible that he intended to keep Hamlet in Elsinore so that he could plot to get rid of him.
Hamlet doubts the ghost's statement that Claudius had murdered him--the ghost could be lying. He re-enacts the murder in a play and has it performed in front of Claudius. Sure enough, it causes an attack of conscience in Claudius.