I would like to add a comment to the fine answer above.
Achilles displays arrogance whenever he has the opportunity--although only a degree away from other warriors' arrogance--and he certainly mistreats Hector's corpse, an action deprecated by Homer as excessive even by Greek warrior-culture standards. Other Greek warriors, however, do not view Achilles' vengeance as particularly excessive because their warrior culture called for them to taunt enemies before battle and for the victors to taunt the losers' corpses after defeat. In fact, in Book 22, after Hector is killed, the other Greek warriors approach his body and repeatedly pierce it.
Although the gods are outraged by Achilles' treatment of Hector's body, their solution is to send Thetis, Achilles' mother, to convince him to give Hector's body back to his father, King Priam of Troy. At this point in the poem, Achilles and Hector share a moment of sorrow over the death of Hector and Patroclus, and Achilles achieves true empathy for Priam and his loss.
Achilles, who dies after the period covered by the Iliad, is aware that he is fated to die after the death of Hector (his mother has foretold it), but Nemesis, the personification of righteous anger, is not involved in Achilles' death, which is his allotted fate from his birth, not the result of retribution for his actions.
Nemesis is one of the Greek goddesses of fate. She is particularly concerned with carrying out retribution against those who have undeserved good fate, who misuse power, or who otherwise abuse good fortune and become arrogant. Although she is not a major figure in Homer's Iliad, the concept of nemesis is applicable to several characters in the story.
The first character is Paris, who has returned to Troy and is living happily in an adulterous relationship with another man's wife. He is also a coward, by Greek standards, preferring to fight at a distance with a bow and fleeing when challenged to a sword fight. His death, and Oenone's refusal to prevent it, would be an example of nemesis.
Similarly, Achilles, because he is invulnerable except for his heel, is also arrogant, and mistreats Hector's corpse. Achilles is killed by an arrow aimed at his vulnerable heel, another example of nemesis.
Troy as a whole, because of its support for Paris' unjust taking of Helen from her lawful husband, is also destroyed.