Mold around doorknob?
I would temporarily remove the door latch hardware (i.e. door knob and deadbolt) and then use a hand sanding operation to remove the old layers of paint and black stains (i.e. mold and old hand oil prints) down to clean base material.
As you have noted the staining is very good at telegraphing through paint layers and so the only real way to remove all of that completely.
You will want to use care with keeping the sand paper grit fairly fine at the last stages as you get to the base material. This will apply to both vinyl clad and exposed wood frame doors. After sanding it would be a good idea to clean the area with a bleach solution. After dry paint the area with a good quality primer. Use care with the application of the paint to blend the edges up to the unsanded areas to ensure that there is no visible ridge at the paint edge.