How to remove/clean black mold?
This is a problem best suited to the professionals, but if you must do it yourself the number one priority is the safety of you and those around you.  This can be a very hazardous task, if you do not take every precaution to contain and remove the spores properly. 

The first thing you'll have to think about is containment, you don't want to spread the mold spores while you clean the area.  You'll want to seal off the area (under negative pressure) with plastic sheeting, making sure all seams are properly sealed.

Once the area is sealed off, you'll need the appropriate clothing. 


Respirator
Tyvek suit (or other clothing that covers your whole body).
Rubber gloves.
Goggles.


Now it's time to start cleaning. Before you begin make sure the infected area is damp, this will minimize the chances the spores will become airborne. You'll start by scrubbing the affected area with a Borax-water mix (1cup Borax/1 gallon water) using a scrub brush, to remove as much of the mold as possible. Wipe up any excess mess with rags, and dispose of them in a thick plastic bag (Contractor bag). Rinse the area with a small amount of the Borax solution, then let it dry completely.  Once dry you'll want to pour a small amount of disinfectant (Isopropyl Alcohol), and allow that to dry as well.

Now that the mold has been removed, you can paint the affected area with Kilz if you like. The most important step at this point is to address the moisture problem in the area, or it's likely the infestation will return. You'll want to do what you can to make sure there is no standing water in the area, and that any moisture drains away and dries quickly.