His mother was the nymph Thetis, and his father, Peleus, was the king of the Myrmidons. Achillesâ most notable feat during the Trojan War was the slaying of the Trojan hero Hector outside the gates of Troy.Although the death of Achilles is not presented in the Iliad, other sources concur that he was killed near the end of the Trojan War by Paris, who shot him in the heel with an arrow.chilles was the son of the Nereid Thetis and Peleus, the king of the Myrmidons. Zeus and Poseidon had been rivals for the hand of Thetis until Prometheus, the fore-thinker, warned Zeus of a prophecy that Thetis would bear a son greater than his father.