Are you sure its not coming from the Washer?
It is definitely not coming from the washer.
There is very little in a dryer that could cause this There is no grease its more likely burned lint and the machine would have to be taken apart and cleaned
check the drum seals to make sure they are not loose allowing lint to escape into the dryer and bypassing the filter
if there is lint in the heater it will burn makeing black soot
this could be what you are seeing
without seeing it first hand there is no way to know