Achilles is the leader of the Myrmidons who fight on the side of the Achaeans (a collective term for the Greeks and their allies) against Troy. He is the son of Thetis, a sea goddess, and Peleus, a mortal. He is a pure warrior concerned with honor and glory—fierce and merciless in battle. His rage is born when Agamemnon replaces the loss of one of his prizes of war by seizing one of Achilles's prizes. Until Agamemnon apologizes, Achilles and his men will not fight for the Achaeans. He actually asks the gods to ensure Agamemnon and his forces, his own allies, experience defeat without him, resulting in much bloodshed. His sense of honor doesn't allow for any compromise, but the loss of a friend and the appeals of an enemy bring out his humanity in the end.
Whereas Achilles is a warrior and nothing else, Hector is more multifaceted. He is also a brave and fierce warrior, but as a prince of Troy, he is fighting for his city and family as much as for honor and glory. His interactions with his wife and son, as well as his brothers, demonstrate a strong sense of responsibility for protecting his city and the people in his family. Unfortunately, his duty to Troy gets overridden by his pride, leading him to foolishly face Achilles in a fight he cannot win.
Agamemnon, a powerful and proud man, is the top commander of the Achaean army. He feels entitled to take whatever prizes and honors he wants without apology. When he tries this with Achilles, he incurs the great hero's rage, with dire consequences for the whole army. Even after he realizes the costs of his actions, he offers only reparations, not apologies. He is a good fighter, but his leadership style is harsh, and he shows no mercy to enemies. He has been promised victory at Troy, but he repeatedly has to be talked out of fleeing when the battle goes against the Achaeans. Agamemnon is often called "Atrides," meaning "son of Atreus" (as is his brother Menelaus).
King Priam presides over the city of Troy and daily life while his many sons fight off the attackers. He shows compassion and understanding to Helen although she brought ten years of war upon his city, and he reasons with his wife, Hecuba, in her distress. Unlike the warriors, he is not ruled by pride. He humbles himself to appeal to Achilles personally for the return of his son's body, an act that finally brings out Achilles's human side.
Achilles's sea-goddess mother, Thetis, protects and advocates for him throughout the poem. She carries his angry wishes to Zeus and persuades him to grant them, setting up the many defeats that the Achaeans suffer through most of the poem. She also makes sure her son is protected, replacing his lost armor with a new set forged by a god. She is tortured throughout the poem by the knowledge that her son is fated to die.
Paris (also called Alexander in some translations) committed a daring act in stealing Helen away from Menelaus, but he is more a lover than a fighter at heart. He can be roused to honorable combat, but he is not terribly effective at it and often prefers to stay in his rooms with Helen. In contrast to Helen, who deeply regrets the consequences of her actions, Paris doesn't seem to feel much responsibility for the ten years of war he has brought upon his city. However, his status as a prince and the favor of the goddess Aphrodite make him one of the heroes of Troy. (After the end of The Iliad, he is the one who kills Achilles.)
Although Achilles is his king and commander, Patroclus is older and wiser. They were raised together, making them nearly brothers. Patroclus is much more compassionate than Achilles, and he feels the suffering of their fellow Achaeans. When he tries to temper the effects of his friend's immovable rage, he succeeds only in bringing about his own tragic death. Patroclus seems to attract great loyalty and love from others. The bond between Achilles and Patroclus is very close—as close as self and shadow. Likewise, Briseis laments the death of Patroclus nearly on the scale of a bereaved wife (Book 19).