Hi WS and welcome to the forum.
If you have the luxury of moving the stove out of doors, as I have done in the past, a high-powered leaf blower shoved in the rear outlet does a wonferful job of getting rid of the flyash build-up. Remove the ash pan and blow out the air intake duct as well.
During the season, I use duct tape to attach a length of garden hose to my shop vac and snake it into the left and right exhaust ports. It gets a lot of the ash that builds up in the back corner. If I disconnect the stove pipe from the rear outlet, I approch it from that side too and get some more ash out. I have also used a wire coat hanger to scrape out the ash.
The last time I purchased a used stove to recondition, the fellow included a wooden handled brush on a long flexible wire cable. It appeared to be a great tool for cleaning out the ports but has since disappeared into the black hole that is my garage and I will probably never see it again.