If you've read The Iliad....?

Achilles kills Hector to avenge Patroclus, but I wouldn't say that his revenge is very good medicine, since it doesn't do much to ease his grief, and he knows that he will soon have to pay for it with his own life. He's warned twice that he's fated to die soon, maybe even next, after Hector. In fact, although Homer never says so explicitly, Achilles knows in his heart that HE is responsible for the death of his friend Patroclus. When he withdrew from fighting because Agamemnon took his captive woman from him, he prayed to Zeus to let the other side win for a while so that his own side would see how badly it needed him. Then, as his prayer was being granted, he let Patroclus go into battle in his place and his armor. So when Hector kills Patroclus, Achilles must realize that he is the one really responsible. But that's a very uncomfortable thought, so he projects all his anger onto Hector, and when, as he prepares to return to battle, he receives that warning, he shrugs it off. Evidently he feels that he owes it to Patroclus to avenge him even at the cost of his own life. The fact that Hector faces Achilles wearing Achilles' old armor, which he took off the body of Patroclus, underlines what Achilles is really doing in killing Hector.