What was Achilles's reaction to Patroclus's death?
In The Iliad, by Homer, Achilles refuses to fight for the Greeks in the siege of Troy after King Agamemnon takes his concubine. Achilles' refusal turns the tide of war against the Greeks. To prevent seemingly imminent defeat, Achilles' best friend (and, in some readings, his lover), Patroclus, begs Achilles to let him use Achilles' famous armor in battle. Achilles reluctantly lets him use it, and Patroclus staves off defeat before dying in battle with Hector.
Achilles' reaction to the news of Patroclus' death is intense. He is mad with grief, and vows to kill Hector, despite knowing that the death of Hector is destined to bring about his own demise. His mother, the goddess Thetis, brings him a new set of armor, newly made by the god Hephaestus, despite also knowing that Achilles is destined to die soon after the death of Hector. Achilles returns to battle in his new armor, ravages the Trojan army, kills Hector, and defiles Hector's body, which raises the ire of the gods and cements his own demise.
Achilles is absolutely devastated by Patroclus's death. Patroclus was his bosom buddy, his best friend in the whole world. They had such a strong connection, both as friends and soldiers. It's a sign of just how much Patroclus meant to Achilles that his friend's death causes him to return to battle.
Previously, the Achaeans' finest warrior had been sulking in his tent, furious at being disrespected by Agamemnon. No matter how much his comrades pleaded with him, no matter how many Achaeans were slaughtered by the Trojans, Achilles steadfastly refused to set foot outside his tent and give battle. It took the death of his closest friend to get him to come out.
For Achilles, the death of Patroclus is personal; he regards the actions of Hector in killing Patroclus as an insult to him. So even when he finally returns to the fray, Achilles is fighting to avenge the death of his fallen friend and comrade, not for the greater glory of the Achaeans in their seemingly never-ending war with the Trojans.
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.