How should I treat mold under the caulk around my kitchen sink?
Never caulk an overmount sink into place.  If you've done it right, you will have to destroy the countertop to get it out if you have to remove it down the road.  Silicone is incredibly strong when used correctly.  

First, remove the sink.  Then spray bleach and water (50% v/v) all around the mounting flange and the hole in the counter.  Let it sit for a few hours and repeat 2 more times for a total of 3 times.  

Second, get yourself some stainless plumbers putty.  Roll it into thick ropes (1/2" dia) and place it all around the bottom side of the mounting flange. It should form a continuous ring underneath the lip of the sink. Just join the ends together by squishing them into each other. 

Third, reinstall the sink and tighten the mounting screws a little at a time all around the perimeter of the sink.  Trim the excess putty away from the sink flange on the top of the counter once it's done.