Why does Caludious want Rosencrantz and Guildenstern to spy on Hamlet?

Claudius wants Rosencrantz & Guildenstern to try to find out whether Hamlet is a physical danger to Claudius. Claudius is worried Hamlet might try to get the crown the same way he did, by murder. Since R & G are Hamlet's old friends, Claudius thinks Hamlet might reveal his ideas about that to them.

This is why R & G speak of "ambition" when they talk to Hamlet. If a Prince is ambitious there's only one step up for him, and that's to become King.