1. To get rid of Hamlet: Claudius fears Hamlet's growing suspicion and potential for rebellion. He believes Hamlet is a threat to his power, especially after Hamlet's public display of grief and his thinly veiled accusations of murder. Sending Hamlet away, ostensibly on a diplomatic mission, allows Claudius to distance himself from the young prince and potentially eliminate him altogether.
2. To have Hamlet killed: Claudius secretly writes a letter to the King of England, requesting that Hamlet be executed. He uses this as an opportunity to get rid of Hamlet permanently, removing any threat to his reign. However, Hamlet discovers the plan and alters the letter, ensuring his own survival and the death of Rosencrantz and Guildenstern, the two courtiers sent to accompany him.
It's important to note that these reasons are intertwined. Claudius uses the pretense of diplomacy to achieve his true aim, which is to eliminate Hamlet.