Achilleus was the son of Peleus, king of the Myrmidons in Thessaly and all-around hero, and the sea nymph Thetis. Zeus and Poseidon had vied for her hand until an oracle was revealed that she would bear a son greater than its father, whence they wisely decided to give her to someone else. According to the usual legend Thetis had tried to make Achilleus invincible by dipping him in the river Styx, but forgot to wet the heel she was holding him by, so he could be killed by a blow to that heel. Homer, however, deliberately makes no mention of this; Achilleus can not be a hero if he is not at risk.
There was an oracle stating that if Achilleus fought at Troy, he would die there. His mother thus hid him at the court of Lycurgus in Scyros, disguised as a woman. There he had an affair with Deidamia resulting in a child, Neoptolemus. He was found out, however, by OdyssEus?, who arrived disguised as a peddler with trinkets and weaponry. Achilleus was marked out from the other women by admiring the wrong goods. Alternatively, he was found out by a blast of the trumpet, whence instead of cowering he grabbed a spear to ward off the attackers. From there he needed little convincing to go to Troy.
Achilleus is one of the only two people described as "god-like" in the Iliad. This does not just refer to his supreme fighting ability, but also to his attitude. He shows a complete and total devotion to the excellence of his craft and, like a god, has almost no regard for life. Not his own - clearly he does not mind a swift death, so long as it is glorious - and not really of others. His anger is absolute. The humanization of Achilleus by the events of the war is the main theme of TheIliad, and is best described on its page.
Shortly after the death of HecTor, Achilleus defeated Memnon of Ethiopia and was very soon after killed by Paris - either by an arrow to the heel, or in an older version by a knife to the back while visiting a Trojan princess. Both versions conspicuously deny the killer any sort of valor, and Achilleus remains undefeated on the battlefield. His bones are mingled with those of Patroclus, and funeral games are held. His armor leads to the death of Ajax.
In TheOdyssey?, there is a passage where OdysEus? sails to the underworld and converses with the shades. One of these is AchillEus, who greeted as "blessed in life, blessed in death", responds that he would rather be a slave then be dead. This has been interpreted as a rejection of his warrior life, but also as indignity to his martyrdom being slighted.
The kings of Epirus claimed to be descended from AchillEus through his son. Since AlexanderTheGreat's mother was an Epiran princess, he could also claim this descent, and in many ways strived to be like his great ancestor. He is said to have visited his tomb while passing Troy.