Pieces of soft black gunk on dishes
It looks like mold or black mildew. 

It commonly starts growing on and in the door gasket, which is somewhat porous. It collects on the filters until, as you have learned, you clean them. Then small pieces escape and stick everywhere. 

Bob Villa and others recommend bleach or vinegar for controlling mold in a dishwasher. But your infestation looks too far advanced for such mild treatment. 

You should use a cleaner called Affresh which was designed by front load washer manufacturers to remove mold and mildew build up. 

Unfortunately Affresh and like products are not good at cleaning the mold from porous rubber parts. Try flexing the gasket away from the tub and scrubbing behind it with a stiff toothbrush.

If this doesn't work then you will probably have to replace the door gasket. The part costs around $80.