How should I clean the dishwasher filter

Monthly vinegar use: a full cup in an empty washer on "pot-and-pans" mode with no heated dry.

This has kept my unit functioning well and lime/scale free since I tore it apart and discovered the pump, the "macerator", pretty much the whole lower works, plugged up with lime, scale, and a whole lot of solidified dishwashing soap. I predict that you will be disassembling the lower works, internet searches helped me tear mine down (along with trial and error and persistence) and put it back together.

I will never use bargain liquid or any type of granular soap again. I only use the gel pouches and it's been smooth sailing ever since.

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.