Patroclus convinced Achilles to let him lead the Myrmidons into battle. He wore Achilles’ armor and pushed the Trojans away from the Greek ships and back toward Troy. Hector eventually kills him which is what bring Achilles back into war.
The Road Back
Achilles hears that Hector has killed Patroclus and is determined to kill Hector. He and Agamemnon end their quarrel, and Achilles shows his strategies on the battlefield. As Hector flees from him, Achilles chases him around the walls of Troy before finally killing him.
Paris killed Achilles at the Scaean Gates by shooting an arrow into his heel. The Greeks burned the body of Achilles and placed his bones with those of Patroclus. Despite this, Thetis rescued Achilles from the funeral pyre and took him to White Island in the Euxine Sea, where can be immortal.
Return with Elixir
Achilles himself does not experience an actual road back as he was dead. However, his road back in a sense, could be the idea of dying and meeting the afterlife.
Obviously before Achilles' death, the Trojans mourned Hector’s death and were angered that Achilles wouldn't return his corpse. The gods then guided Achilles to accept Priam’s (father of Hector) ransom and surrender Hector’s bod in which he did not want to let go of.
Achilles is remembered from that point as the first tragic hero in literature. he sacrificed his life to eventually achieve that everlasting fame via his legacy.