Achilles' stubborn and cowardly sides are shown when Ajax, Phoenix, and Odysseus travel to visit him and beg him to return to the fight. Instead of acting like an honorable warrior, he refuses after listening them plead for his help in the fight. He said that he would not return until his own Greek warriors' ships were on fire from Trojan flames.
Achilles' rage and childishness is revealed when he learns of his friend, and father figure's death, and becomes not only enraged but cries like a child. Children, when they get angry because of something that someone did to them, their first thoughts of what they can do to get back at them, to make things "even." This is what Achilles does. When Hector killed Patroclus he also took the armor that Patroclus was wearing which belonged to Achilles. This is the point at which Achilles' values show because this when he decides to return to the fight, to avenge his friend's death.
As part of his "revenge" Achilles kills every Trojan he sees without mercy to get to Hector. He forces the Trojans to retreat all the way into their own walled city. This is where Achilles and Hector meet to avenge Patroclus' death.
Hector is beat by Achilles and asks him for mercy in his death. Achilles refuses, also being consumed by his ego he doesn't act as the better person despite what his previous experiences have taught him about how to act. So he tortures Hector before...