You may have heard that a bleach mixture is ineffectual for dealing with mold in an affected area. Actually, it can be ineffective on porous surfaces. But in the bathroom, you are dealing with the caulk line, which is non-permeable, and the non-porous surfaces in your shower.
So, using bleach and cotton balls kills mold just fine as long as you wear gloves and other protective gear like eye protection.
But if you do not want to use this method or if it is not getting the job done for whatever reason and there is visible black mold, there are some other ideas you can try.