How does Claudius manipulate Laertes?
When Laertes returns to Denmark, he seeks vengeance for the death of his father, Polonius. Laertes blames Claudius. Claudius sees this as an opportunity to protect himself against Laertes and to make an ally of Laertes against Hamlet. Claudius swears he has nothing to do with Polonius' death. He tells Laertes he sent Hamlet away because of Gertrude's love for Hamlet (and because he couldn't outright accuse Hamlet because he is beloved by the people.) Claudius is thus inferring, to Laertes, that he sent Hamlet away because Hamlet is responsible for all of this: Ophelia's madness, Polonius' death. Hamlet is responsible for his mistake of killing Polonius and Ophelia's madness is a consequence of Hamlet's intentional (methodical) madness, but it is Claudius how had set all these things in motion with the initial murder.
Claudius speaks of his plot to kill Hamlet and Laertes agrees to take part. Claudius lures Laertes into his plot to get rid of Hamlet and by making Laertes an ally against Hamlet, Claudius also is trying to protect his crown from Laertes himself.