Achilles is the son of Peleus and Thetis.
When Thetis gave birth to Achilles, she decided to make him invulnerable to weapons. She took him to the entrance of the underworld, and dipped him into the river Styx. As she was about to fully submerse her child, she was startled by Peleus. He must have feared losing yet another son, and followed his wife to the underworld. Because of this, Thetis did not completely dip son in the river, leaving the young Achilles' heel still vulnerable.
Peleus did not raise the child on his own at first, but instead gave him to the centaur Chiron. He taught the young Achilles to fight and run like a deer. Then, Achilles returned to Peleus' court, where he was trained further by Phoenix. It was during this time that Achilles met Patroclus, who had fled from Opus. The two became lovers.
When the Trojan War began, Thetis feared for her son. She knew that he would die an early death in Troy, and attempted to defy the Fates. She took him to the court of Lycomedes, on the island of Scyrus. She persuaded Lycomedes to take her son in, and to raise him as a girl. During his time here, he has a son with a daughter of Lycomedes. The child's name is Neoptolemus.
This ruse might have worked, but the seer Calchas predicted that the Greeks would not win, unless the young Achilles fought. The Greeks eventually came to Lycomedes' court, and Odysseus was not fooled by this ruse. He had one of his soldiers sound the call to war, and Achilles immediately ripped off his clothes and grabbed some nearby armor. Now discovered, Achilles was forced to sail for Troy. With his fifty Myrmidian ships and friend Patroclus, Achilles accompanied Odysseus to Asia.
For the first nine years of war, the Greeks concentrated on taking the towns surrounding Troy. Achilles, although still very young, seems to already by this time have distinguished himself as a warrior. In the Iliad, he is said to have sacked twelve cities by sea, and eleven by land.
In the tenth year of the war, Agamemnon was forced to return his concubine Chryseis, in order to end a plague. As compensation, he took Achilles' concubine Briseis. His pride hurt, Achilles promises to not fight again for the Greeks. Achilles then prays to his mother Thetis for help. He asks her to go to Zeus and beg him to help the Trojan cause. In this way, when the Greeks are pinned against their ships, Agamemnon will clearly see the mistake he has made. Thetis agrees to do as her son has asked.
For a time, the Greeks hold their own. Eventually, with Zeus's aid, the Trojans begin to push back them back. Hector and the Trojans slowly begin to gain the momentum. Seeing the fortune of his forces, Agamemnon, under the advice of Nestor, sends Odysseus, Ajax of Salamis and Phoenix to Achilles' tent. They entreat Achilles to return to battle. He offers to return the girl Briseis, as well as a handsome reward. Achilles pride has been hurt too deeply, however, and does not accept.
He is, however, worried that the Greeks may lose. He allows his friend Patroclus, while wearing his armor, to lead the Myrmidons back into the battle. When the Trojans see Patroclus, they think Achilles has returned. The Trojans panic and flee. Patroclus, however, is soon killed by the Trojan Hector, and his armor stripped.
When he learns of his friend's death, Achilles, now filled with rage, returns to the fight unarmed. The Trojans turn and flee at his sight. Achilles' mother Thetis comes down and gives her son new armor, forged by Hephaestus himself. Achilles then chases them from behind, preying on the slow. As the other Trojans flee into Troy, Hector remains alone outside the walls to fight.
In their duel, Achilles spears Hector in the neck. He then taunts the dying man, saying that he will not be buried. Struggling for breath, Hector foretells Achilles' death at the hands of Paris and Apollo, and then dies. Achilles then strips the body and ties it to his chariot. Day after day, he drags the body around the walls of Troy as revenge for the death of Patroclus.
After twelve days, the gods could take no more. Thetis was ordered to convince Achilles to return the body. Hermes went to Troy, and accompanied Priam to Achilles' tent. The old man arrived, crying and begging for his son's body. Achilles relented from his rage and agreed to return the body. He even agrees to keep the Greeks from fighting for ten days, so that Hector may be buried properly.
Soon after, however, Achilles' life was ended, as had been prophesized many years earlier. From the walls of Troy, Paris shot an arrow, which, guided by Apollo, struck the heel of Achilles and killed him.