In some respects, Homer's Iliad is a long meditation on why one fights. The epic describes gods and men at war with each other, each one perhaps having a different motive for doing battle with the others.
Achilles entered the Trojan War not for any reason of personal grievance but to gain glory for himself. He was by far the Greek's greatest warrior, and it was believed that the Greeks could not win Troy without him on their side. However, Agamemnon insults Achilles by claiming his war prize, Briseis, a woman whom Patroclus promised Achilles would marry. This begins the dissent between Agamemnon and Achilles that causes Achilles to sit out part of the war, much to the disadvantage of the Greeks.
Patroclus is a disposable character, and he serves at first to soften Achilles, who tends to be arrogant. He also is Achilles's dearest friend and possible lover. In an emotional sense, he is something of an "Achilles's heel," since his pleading to enter battle causes Achilles to act against his better judgment. His death causes Achilles to enter the battle, despite his earlier determination to refrain from fighting for the Greeks.
Patroclus himself is eager to win glory, and he enters the battle in Achilles's armor, ignoring Achilles's warning to not pursue the Trojans to their gate but to only prevent them from advancing on the Greeks. When Apollo intervenes, Patroclus is vulnerable to Trojan attack and is eventually killed by Hector.
In his overwhelming grief, Achilles re-enters battle with a personal motive: to kill Hector in revenge for Patroclus's death. At this point, the battle becomes increasingly personal, as Achilles fights out of anger and grief and for honor and revenge, while Hector seeks to save his city and his family.
When Achilles returns to battle, he fights for honor, not glory; he fights for the love he bore Patroclus, not the admiration he hoped to gain from others.