Heavy build up in your rim sounds nasty....
Are you talking about carbon rings in chamber? If so, many solvents will struggle in my experience.
Soaking in kroil (if you can get it) is good but dont go soaking with ammonia based solvents. Im currently polishing a rimfire chamber and its bloody hard work having found a monster of a carbon ring. Previous owner deserves a bullet for such negligence hahaha
KG1 is fast and easy