Chlorine kills bacteria so you shouldn't have any problems with the chlorine, unless you are allergic to it. It could actually be good for the piercing. But the ring you are using could become discolored if it isn't a pure metal.
If your piercing is new, you should avoid swimming until it is fully healed. If you are healed, it's the same as caring for pierced ears. Rinse them and dry!
If your naval piercing is adequately healed, the chlorine should have no effect on it. If your piercing is new, I would wait until it's properly healed before exposing it to chlorine.
If the piercing is completely healed, and that can take upwards of six months, then swimming in chlorinated water should be fine. If it has not healed, it may damage the skin or scab.
If you're concerned about swimming with the piercing, don't be. Just clean the area thoroughly after swimming. If the piercing is healed, remove the ring and clean it with peroxide.
