The Iliad ends with the burial of Hector. There is nothing about the death of Achilles, only predicted by Hector. In Greek Mythology did he really die? Or was it added in Roman Mythology together with his weak spot on the heel and the whole "bathing in the Styx stuff"?
The death of Achilles is related in Aethiopis by Arctinus of Miletus in the 8th Century BCE. This is one of the poems the Ancient Greeks collected into an "Epic Cycle" centered around the events of the Trojan War. Like most of the Epic Cycle except Iliad and Odyssey, Aethiopis is mostly lost, except for five lines and second-century CE summary (in Greek) by Proclus.
The Cypria, described in the preceding book, has its sequel in the Iliad of Homer, which is followed in turn by the five books of the Aethiopis, the work of Arctinus of Miletus. Their contents are as follows. The Amazon Penthesileia, the daughter of Ares and of Thracian race, comes to aid the Trojans, and after showing great prowess, is killed by Achilles and buried by the Trojans. Achilles then slays Thersites for abusing and reviling him for his supposed love for Penthesileia. As a result a dispute arises amongst the Achaeans over the killing of Thersites, and Achilles sails to Lesbos and after sacrificing to Apollo, Artemis, and Leto, is purified by Odysseus from bloodshed. Then Memnon, the son of Eos, wearing armour made by Hephaestus, comes to help the Trojans, and Thetis tells her son about Memnon. A battle takes place in which Antilochus is slain by Memnon and Memnon by Achilles. Eos then obtains of Zeus and bestows upon her son immortality; but Achilles routs the Trojans, and, rushing into the city with them, is killed by Paris and Apollo. A great struggle for the body then follows, Aias taking up the body and carrying it to the ships, while Odysseus drives off the Trojans behind. The Achaeans then bury Antilochus and lay out the body of Achilles, while Thetis, arriving with the Muses and her sisters, bewails her son, whom she afterwards catches away from the pyre and transports to the White Island. After this, the Achaeans pile him a cairn and hold games in his honour. Lastly a dispute arises between Odysseus and Aias over the arms of Achilles.
Proclus, Chrestomathia, from the Loeb Classic Library, translated by Hugh G. Evelyn-White, available at the Theoi Project.