Why did Achilles cry over Hector's body?

It's a movie so we can all only guess. The only clue we have, that after Achilles wept, he said:

We'll meet soon my brother.

My explanation is this:

Achilles was thirsty for revenge and he already fulfiled this need. When Priam visits him, it completely changes his perspective. He admires bravery, and honor. He despises the king he serves to, Agamemnon. He probably saw that Hector had what he never had - a honorable king and father, honorable purpose to serve to. He was clearly deeply touched by the values Hector and Priam had shown, the values he admires so much. Suddenly, he had nothing but admiration for them. He saw Hector as brother, because they shared together his most important values. And he killed him. Achilles could have probably felt like he killed himself, for a reason that might seem absurd from that perspective... That's why the tears and that's why he said "we'll meet soon my brother".

This isn't in the source material, Homer's Illiad. It was sort of a "bastardization" of Achilles' character, as he grieved for his own father and the fact he knew he'd never see him again as he was destined to die in Troy, as well as his good friend Patroclus. I believe in the movie it was used to show his admiration of Hector as a worthy opponent. In Homer's original story, Achilles and Hector were sort of a Yin/Yang representing the best of Greece and Troy with dignity, honor and bravery.