How and why does Claudius manipulate Laertes in Act 4 of Hamlet?
Claudius does feel guilty enough for killing his own brother that he cannot bring himself to kill his nephew/stepson as well. Besides, having two members of the royal family die so quickly might call undue attention to the surviving family (shades of Macbeth).
Claudius sees that Laertes is a "man of action" and chooses to take advantage of his anger and direct it towards Hamlet. While Hamlet is indeed guilty of stabbing Polonius and hiding the body, it was Claudius' scheming that put Polonius behind the arras.
Hamlet also indirectly caused Ophelia's insanity, because she was so dependent on the men in her life (Polonius, Laertes, and Hamlet) and they have all abandoned her. It was easy for Claudius to direct Laertes' anger toward Hamlet.