After every use, I clean all the ashes out of the firebox and brush/wipe down the grates and deflector plates. Clean all grease, etc. from the bottom of the cook chamber.
When I notice deposits inside or at the opening for the stack, cook chamber lid, cook chamber walls, I brush them down and clean the deposits out. On my COS Ok Joe, I'll remove the stack and scrub the inside with a wire brush attached to a wooden dowel. That way I can get all the deposits into a bucket.
That's about as far as I go with cleaning...