Why did Achilles return to the Greeks?

Answer 1

The death of his best friend, Patroclus, caused Achilles to fight again as Patroclus was coaxed into going into battle with Achilles armour and Hector killed Patroclus, beliving that it was Achilles.

Answer 2

I'm not entirely sure which time you mean, so I'll put both.

The first time (when he leaves hiding to go to Troy): Achilles joins the Greeks because he wishes to pursue glory, and although he knows that he is destined to die in battle, he wants to be remembered and revered.

The second time (when he returns to battle after having pulled the Myrmidons and himself from battle after losing Briseis): Achilles rejoins the battle because his friend, Patroclus, is killed by Hector. Seeing as Patroclus was very close to him, Achilles seeks revenge for his death.