Vinegar works better as it kills the mold spores, bleach does not.
You can just use a rag, the bleach will help with the staining and taking away the surface mold, but it will not kill the mold spores. After finding and drying the area, spray on vinegar to kill the spores everywhere on the caulking. It is possible if the caulking has been on the tub for awhile it may have gotten into the edges. If this is the case, you would be better off removing and replacing the caulking and just start fresh. Clean the area with the bleach water, spray with the vinegar and let dry thoroughly before you put on the new caulking.
Agree with first bleach to help with staining and vinegar to kill the spores!