I deal with mold on a daily basis at Titan Restoration. And as per my experience, the first thing you need to do is fix the moisture problem. It is very much possible that the mold is being fed by a water leak on the other side.
There are many ways to get rid of mold which use everyday household products that you probably already have like bleach, borax, vinegar, hydrogen peroxide, baking soda, etc. Spray or apply it to the mold. Wait for 10-15 minutes and wipe with a rag or scrub appropriately.
However, if the mold spreads over a larger area, it's always better to call in a professional. They have advanced training and techniques to completely eliminate the mold infestation and restore the property without risk of further growth or dispersal.