Patroklos and Achilleus have the world's best bromance. They adore each other. They have each others' backs. And this, of course, makes it suck when Patroklos is killed.
Patroklos has been Achilleus's best friend since childhood. Even though Patroklos is older than Achilleus, he is much weaker. When the two of them are sailing off for Troy, Patroklos's father Menoitios reminds Patroklos that his most important role is as an advisor to his friend. We don't see Patroklos doing this very often; most of the time, he simply hangs out with Achilleus, accompanies him on raids, and gets stuck performing minor chores—like checking on the identity of Machaon, the healer.
When Patroklos does advise Achilleus, it is usually by acting as his conscience. We get a sense of this from Briseis's speech in Book 19, when she tells us that Patroklos promised to convince Achilleus to take her as his lawful wife. We also see it when Patroklos convinces Achilleus to let him lead the Myrmidons into battle. His tears—which Achilleus mocks—are an obvious sign that he is acting out of pity for the other Achaians.
In the battle, Patroklos shows himself to be a courageous and effective warrior. This can be seen when he kills the Trojan ally Sarpedon. That said, once Achilleus has let him fight the Trojans, Patroklos succumbs to over-enthusiasm. Disregarding Achilleus's instructions not to press on toward the city, he even goes so far as to make repeated attacks on the city's walls, each of which is repulsed by the god Apollo. (He smartens up when the god tells him to back off—that fate won't even let Achilleus take the city.) Shortly afterward, he dies at the combined hands of Apollo, Euphorbos, and Hektor.
With his dying breath, Patroklos preserves his sense of pride, telling Hektor (accurately) that he can't claim all the credit for killing him:
"Now is your time for big words, Hektor. Yours is the victory
given by Kronos' son, Zeus, and Apollo, who have subdued me
easily, since they themselves stripped the arms from my shoulders.
Even though twenty such as you had come in against me,
they would all have been broken beneath my spear, and have perished.
No, deadly destiny, with the son of Leto, has killed me,
and of men it was Euphorbos; you are only my third slayer." (16.844.850)
Then, mysteriously, Patroklos prophesies Hektor's death at the hands of Achilleus. This supernatural aura still clings to Patroklos in his final appearance as a ghost in Achilleus's dream—though, of course, it's hard to say if it's really him or not. His final words to Achilleus— in which he asks for their ashes to be buried together—remind us of the depth of their friendship.