Why does Achilles continue fighting after killing Hector?
He only returned to avenge Patroclus and he did that. Didn't he end up feeling empathy for Priam and the Trojans?
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In Madeline Miller's Song of Achilles she posits that, after the death of his beloved friend, Achilles is seeking death.
This makes sense because the prophecy about Achilles' death was that if he took part in the fighting at Troy, he would die there. Once he enters combat to avenge Patroclus, his fate is sealed.
Miller presents his time post-Patroclus as one of inconsolable mourning for his lost companion. Death is a release from this conscious suffering. There is at least the chance of being re-united with Patroclus in the underworld.
At first, Achilles came to the war for glory. Being the conqueror in one of the biggest battle of all times, conquering the city of Troy which was famous for being so impregnable... this was naturally challenging his warrior instincts. The original reluctance to fight along with Agamemnon disappeared after Patrocles died. He could now finish the campaign and be the hero to conquer Troy.