Trojan War: Achilles vs Hector

Ok recap:
Suitor’s military alliance: Helen’s “kidnapping” sparked a war with Troy and her allies. Leaders of Greece assemble for the Rescue of Helen and the Destruction of Troy.
Death of Lover: Patroclus, Lover of Achilles, was killed by Hector and unleash the fury of Achilles

To War:

Achilles came to the battlefield, with new armor and shield from his mother, fashioned by the god Hephaestus. Homer gave a long description of the shield's design. Achilles was determined to seek out and kill Hector in single combat.
Zeus decreed that the Olympians were allowed to visit the battlefield, now that he had fulfilled his vow to the goddess Thetis. Another reason that the other gods was allowed to re-enter the battlefield, is to check Achilles from sacking Troy today. Achilles was not allowed to capture Troy; the city would not fall until its appointed time. This seemed to indicate that Achilles could change the course of history if his murderous rage was left unchecked.

Achilles set out to avenge his boyfriend, killing many Trojans and driving them back towards city in defeat. Poseidon, who normally favored the Greeks, saved Aeneas from Achilles. Poseidon told the Trojan hero that he was destined to later ;).
Hector faced Achilles alone in single combat.

Though Zeus admired Hector for his courage and piety, the god could not save him, since the Trojan hero was destined to die that very day.

For twelve days, Achilles mourned over the loss of his companion. Patroclus' spirit came to Achilles in a dream, begging the hero to bury him. A funeral was held in the morning. Twelve Trojan captives were sacrificed at the funeral. Great funeral games were held in Patroclus' honor. Achilles presided over the games, while other Greek leaders took part in the events, such as wrestling, boxing, archery, a footrace and a chariot race.

The night after the funeral gods demand that Achilles return Hector. With the aid Hermes that Priam managed to slip into Greek camp and entered Achilles' tent without letting the other Greek leaders know of his presence. Achilles dared not to disobey the gods, made ready to accept the ransom. Achilles didn't expect that Priam himself would come to his tent in person and alone.

The aged king pleaded Achilles to remember that his own father would be the same age as him and that Peleus would also be concern with his son's well being. Achilles treated Priam with great respect and chivalry. Achilles knew that his own death would follow shortly after Hector, but Troy was doomed to fall not long after, as Achilles' mother had foretold. Achilles even granted Priam twelve days respite from fighting, until the funeral of Hector was completed.

Priam brought Hector's body back to Troy, where a great funeral was prepared. Even Helen mourned for him, since he had always treated her with respect, whereas everyone else treated her with contempt since she was one of the causes of the war.

On the Twelve day; the Trojans received further aids, first from the Amazons and later Ethiopians. The Amazon was lead by Queen Penthesilea  a daughter of Ares. When she arrived at Troy, she boasts of her prowess in battle. But Andromache, the newly-grieving widow of Hector, rebuked the Amazon Queen, asking her to be cautious and not to be boastful, since there are a number of great fighters among the Greeks. After Hector's funeral, the Greeks and the Trojans rejoined fighting out on the Trojan plains.
With Troy's new ally, the Amazons and their Queen drove back the Greeks. Penthesilea killed many Greeks before Achilles killed her. When the Amazon Queen fell, Achilles stripped Penthesilea of her armor, he saw that the woman was young and very beautiful. He seemed to have fallen madly in love with her, and regret killing Penthesilea.

The Trojans received new reinforcement from the Ethiopians the next day. They were led by a prince, named Memnon, son of Eos, the Goddess of Dawn. Memnon killed many Greeks, causing the Achaeans to retreat. In the confusion of the retreat, the aged Nestor was surrounded by enemies, among them was Memnon. Antiochus tried to save his father, but he was killed. Nestor was grief-stricken over his son's death, and tried to confront the Ethiopian prince. Memnon, however, saw no honor in such combat against an old man, so he refused to fight with Nestor. Nestor lamented that he no longer has the strength of his youth. Nestor called upon Achilles to avenge Antiochus and he did.

With Memnon's death; the Trojans lost heart and fled back towards the city's walls, with Achilles in close pursuit. Achilles was at the Trojan’s Main Gate, when an arrow from Paris, guided by the god Apollo, pierced his heel. His heel was the only spot on his body that was vulnerable to weapon (hence the "Achilles' heel"). Personally I think Paris shot him in the ass or mouth with a poisoned arrow. There was fierce fighting over Achilles' body. In the fighting, Ajax killed Glaucus, the last leader of the Lycians. While Ajax carried Achilles' body back to camp, Odysseus kept the Trojans back.
When the funeral was held in the Greek camp, Thetis came with her sisters, the Nereids, mourning over the death of her son. A funeral pyre was lit, cremating his body. His ashes were placed in the same urn, as to that of his beloved friend, Patroclus. Arrangements were made for the funeral games to be held after the funeral. After the funeral came drama; it was decided that the Achilles' armor, made by the god Hephaestus, should be awarded to the best warrior. Ajax and Odysseus both contest for the armor. The Greek leaders awarded the armor to Odysseus. This  pissed Ajax off to the point of homicidal madness. Lucky for the Greek leaders; Ajax killed a herd of sleep instead of them. Ajax commit suicide and receive a decent by Odysseus.