Laertes falls into a bit of a downward spiral following his father’s murder and his sister’s death. He is convinced to work with Claudius to kill Hamlet in a duel and seems to have abandoned his morality. Hamlet did kill his father, so murdering him is an act of legitimate vengeance. Using a sword with a poison tipped dagger in a friendly duel, however, crosses the line and certainly is not keeping in Polonius’ advice to “not then be false to any man.”
Laertes does succeed in killing Hamlet, though it does not seem to be in line with “To thine own self be true.” As Laertes reveals to the audience in an aside, right before killing Hamlet: “And yet tis almost gainst my conscience” (5.2). Laertes is working against his own conscience, in a rigged duel—hardly living up to the standards outlined by his father.
But after the duel, when Hamlet and Laertes are both dying, he does turn a new leaf in an act of redemption, admitting his deceit to Hamlet: “It is here, Hamlet. Hamlet, thou art slain.” He is finally being true to himself and to those around him and even exchanges pardons with Hamlet for their respective murders—a noble last few moments for a dramatic antagonist who was always trying to do the right thing.