Here's a page with details about the Trojan War. A couple of quick differences with the movie:

* Achilles sacked Thebe, east of Mount Ida, killing King Eetion, father of Andromache; and he took Lyrnessus, where he captured Briseis after slaying her husband and not as a priestess of Apollo.
* Paris was at first reluctant to fight, but after accepting Hector's reprimands, he accepted the duel rather than suggesting the fight or being encouraged by Hector.
* Paris fought with Menelaus and got almost killed. But when Menelaus, during the fight, seized him by the horsehair crest of the helmet and began to drag him, Aphrodite came and broke the strap of the helmet, so that it came away empty in Menelaus' hand, and then, to escape Menelaus' renewed attack, the goddess hid Paris in a mist and took him to his own bedroom in the city and not gripping the ankles of his brother Hector....
* And Achilles, who had come to change his mind to some extent, gave Patroclus his armour and sent him to the battle at the front of the Myrmidons, advising him just to remove the Trojans from the Achaean ships, and under no circumstances go in their pursuit rather than running off on his own without Achilles' knowledge.
* Achilles being dead his arms were offered as a prize to the bravest. Ajax and Odysseus competed for them, the latter being preferred by the judges. As a revenge, Ajax planned an attack on the army, but Athena drove him mad, and he slaughtered the cattle with the herdsmen, taking them for the Achaeans. And when he came to his senses he let himself fall upon the sword he had received as a gift from his enemy Hector and died and not being killed by Hector during the first attack on Troy.