I thought the movie was not bad.

Achilles was dipped in the River Styx when he was a baby, and held by his heels, so it was the only part that didn't get submerged. I don't recall if he was specifically shot by arrows, we just learned that he was struck in the heel (somebody betrayed his weak point).
I've never read the books, though. I'm just going by what I learned in mythology lessons.