You're supposed to soak it in warm water with sea salt in it and then a saline solution which is pretty much the same thing. Make sure you don't move the jewelry until it's completely healed otherwise you could get scabs inside. Also make sure you don't touch it at all unless you washed your hands first.
You can use salt to clean it instead, since it helps in healing and can also get rid of infections, lavender oil is also good for use or you can use an antibacterial soap to clean. Avoid using chemicals that are harsh on the sensitive area, like iodine or alcohol.
Dont over clean it. But if you do just in case you should see a doctor. But I would recommend using some type of cleaning method that does not harm the skin..... like for an example alcohol or witch hazel... hope this helps!
NEVER use peroxide on your bellybutton piercing. Use dial soap in the shower, it's cheap and works great, thats what the piercers told me to do 4 years ago, I never had any problems during the healing process because I followed that advice.