The key to this conflict is the fact that Hamlet, in a rash mistake, killed Polonius, thinking that it might have been the King who was hiding behind the arras in his mother's room. Laertes comes storming back to Elsinore demanding vengence for his father's murder. Claudius uses this anger to his advantage. Claudius convinces Leartes to kill Hamlet "accidentally" during a fencing match -- then Claudius's problems with Hamlet will be taken care of but will leave him seemingly blameless in the death. Laertes's anger and frustration are further fueled by Ophelia's madness which Laertes also blames on Hamlet's behavior towards her. Laertes is willing to do anything to avenge his father's death -- he even says he will kill Hamlet in a church! His behavior is much more rash than Hamlet's for the same purpose, but his plans fall back on him as he is killed by the poisoned sword intended for Hamlet.