How do I get rid of mold from the bottom of a bath mat (where the suction cups are)?
My experience is that the black growth actually gets right into the bath mat material and cannot be easily removed. I suggest that you replace the mat and then use an alternate procedure when stowing the mat after use. 

After use always rinse it off well so no bath scum and other things are left on it. Then hang it up over a towel bar, suction cut side up, so that it can fully dry after each use. When you suction cup the wet mat to the wall you are leaving it there as a perfect breeding ground for mold and mildew growth to occur.