I've had my belly button piercing for five months, and around the top hole of the piercing is a purple color but there's no puss coming out.

Skin healing. If there is no pus, no swelling and no tenderness, then the site is probably healing just fine. Some skin heals with a discoloration at first, which fades with time. If the piercing gets pulled on by your clothing during the day, that could be causing ongoing irritation, though, so try not to let it get tugged on.