When you put your indoor / kitchen oven on “self-cleaning” mode, the oven will get as hot as 800° for a very long period of time. This basically incinerates any baked on grease, grime and stuck-on tuna casserole. A wood-fired oven works the same way - except it can get to 1100°+.
If you have any baked on cheese or grease on your hearth fire brick, simply scorch it off with an hour or two of direct wood-fire. This would also be a good time to clean / season any cast iron that may have years of baked on residue. Simply toss that cast iron into the mini-bonfire and it will come out clean as a whistle!
Now - once you have thoroughly incinerated any baked-on gunk, take your brass-bristle brush and push the wood embers to the rear and sides of the oven. Brush the cooking surface w/ a little elbow grease and you should notice that the fire brick is now gunk-free. To get the fire brick hearth cooking surface super-clean, simply take a damp (NOT SOAKING) t-shirt and wrap it around the brush and gently wipe the fire brick / refractory bricks a few times. This will generate a decent amount of steam, but not enough to scaled your face or hands.
Keep in mind, like any self-cleaning oven, you only want to do this cleaning method every-so-often… we only do this 2 or 3 times a year (normal oven usage).