The brief answer to this is that Achilles's reaction to Patroclus's death was one of intense anger and an overwhelming desire for revenge. It was when Patroclus died that Achilles finally got over being angry about having Briseis taken from him. When he heard that Patroclus had died, Achilles stopped "sulking in his tent" and went out to take revenge for Patroclus's death.
Some traditions hold that Achilles and Patroclus were actually lovers and not just good friends. Whether that is "true" or not, Patroclus's death got Achilles angry enough for him to come out and fight. He then killed Hector and many others of the Trojans.
Achilles is enraged when Patroclus is killed by Hector. Achilles will not eat or drink until he exacts revenge upon Hector, even though he knows killing Hector will result in his own death. Achilles' mother tries to help him by having Hephaestus make a new armor for him, but Achilles will die soon anyway. Achilles' anger is so great that he goes on a murderous rampage, killing so many Trojans that the river Scamander is overflowing with the dead bodies of Trojan warriors. Killing Hector isn't enough for Achilles, however; he then drags his body back to the Greek camp in front of Hector's family.