How to remove this black stuff at my countertop corner joint
That looks like mildew behind silicone caulk. I'd scrape it away with a putty knife and razor blade, taking care to not damage the surface. Then use a mild abrasive cleaner to buff away the stain. Consult the surface manufacturer for products and procedures. 

Re-caulk with pure silicone after a good drying interval. Use as small a bead as possible for the best appearance.