Achilles' behavior in the Iliad, at least on first consideration, does not seem to warrant much empathy. Throughout Books 1 through 23, Achilles is, for the most part, a boastful, self-centered, violent warrior who is much more concerned with his own fame than with the Greeks' overall goals in the war against Troy. To be fair, most of the important Greek warrior-kings are concerned chiefly about their fame, but several, Odysseus, Diomedes, Nestor, Idomenous, and even Menelaus, actively assist other warriors who are in trouble. Achilles, aside from his loyalty to Patroclus, is generally a one-man show, and even though his skills and power as a warrior inspire the reader's admiration, perhaps even respect, there is little in Achilles' behavior to elicit empathy--that is, until Hector's death and Priam's attempt to reclaim his son's body.
In Book 24, after Achilles has killed Hector and dragged his body around the walls of Troy, the gods decide that Achilles must give Hector's body back to his father, King Priam, and they send Achilles' mother, Thetis, to tell him that. Achilles coldly agrees. The gods have also told Priam that he needs to approach Achilles himself to ask for the return of Hector's body. When Priam approaches Achilles, he tells Achilles that he and Achilles' father, Peleus, are alike:
. . . think on your own father and have compassion upon me, who am the more pitiable. . . .
Priam is reminding Achilles that Peleus will suffer the same devastation as Priam when he understands that Achilles will not return from the war--just as Priam has suffered the loss of all his sons, most especially Hector. Priam's heartfelt comparison of his Peleus' situation literally stuns Achilles:
He took the old man's hand and moved him gently away. The two wept bitterly--Priam, as he lay at Achilles' feet, weeping for Hector, and Achilles, now for his father and now for Patroclus. . . .
Aside from Achilles' sorrow for the death of Patroclus, this remains the only instance in the Iliad when Achilles steps outside himself to feel the sorrow of another human being. At the conclusion of this scene, Achilles is described as raising Priam up "in pity for his "white hair and beard." Achilles, who has prided himself on his pitiless command of war-making, is finally brought to feel true compassion for someone who is suffering an intense loss.
At this point in the poem, Achilles reaches out to soothe Priam's loss in every way possible. Even though he takes the ransom Priam has brought with him, Achilles treats both Priam and Hector with a respect brought about by Achilles' understanding of someone else's loss, a moment of Achilles' redemption and worthy of our empathy.