How does Claudius manipulate Laertes?
Early in the play, Claudius tries to shame Hamlet for Hamlet's continued grief over the death of his father. He calls Hamlet "unmanly" for his overly developed loyalty and too-deep mourning. After Polonius's death, however, Claudius exploits Laertes's similar feelings of loyalty and grief. He asks Polonius's son,
What would you undertake
To show yourself in deed your father's son
More than in words? (4.7.124-126)
He is prodding Laertes to show his loyalty to his father with more than words, by going after the man responsible for his father's death: Hamlet. When Laertes says that he would "cut [Hamlet's] throat i' th' church," Claudius agrees that there should be no place where Hamlet, a murderer, should find refuge; he says, "Revenge should have no bounds" (4.7.127,129). It's a pretty ironic statement given the fact that Hamlet seeks revenge on Claudius for the death of Hamlet's own father!
Claudius stirs up Laertes's feelings of loyalty to his father, stoking his anger and implying that Laertes isn't really a good son unless he is willing to do more than just talk about his loyalty or his new feelings of ill-will toward Hamlet. Claudius then offers Laertes a dishonorable and sneaky way to avenge his father: by proposing a duel between Laertes and Hamlet and then allowing Laertes to use a blade that has not been blunted for safety. Instead, it will be plenty sharp. Laertes then decides to tip that blade with poison so that even a small scratch will suffice to result in Hamlet's death. By stirring up Laertes's feelings of loyalty for his father and his anger toward Hamlet, Claudius attempts to rid himself of Hamlet without having to get his own hands dirty.
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.