Achilles death is not described in the text of The Iliad. The story of his death is part of later Greek myths. However the death of Achilles is predicted by Hector in Book 22, as he is dying by Achilles' hand. Hector, the prince of Troy, says that Achilles will be killed by Hector's brother Paris (who had stolen or lured Helen away from King Menelaus of Sparta and thus caused the Trojan War). Hector said:
"I know you and what you are, and was sure that I should not move you, for your heart is hard as iron; look to it that I bring not heaven's anger upon you on the day when Paris and Phoebus Apollo, valiant though you be, shall slay you at the Scaen gates." (345)
The later myths tell how Paris shot Achilles, wounding him fatally in the only vulnerable place on his body -- his heel.
There is also a story that Achilles had fallen in love with Polyxena, a princess of Troy, and had gone to the Temple of Apollo to negotiate peace with the Trojans so he could marry her. At this meeting the Trojans betrayed him, and Paris killed him (helped by Apollo.)
None of these stories appear in either of Homer's poems, but in The Odyssey Book 11 Odysseus meets and talks with the ghost of Achilles, so by the time of Odysseus' journey the hero is dead.
Source: Homer, The Iliad. Samuel Butler, trans. Roslyn, New York: Walter J. Black, Inc., 1942.
Paris kills him