Start with a gun cleaner that dissolves carbon. Some of the traditional oil-based cleaners aren't so good at that job. I like Hoppes #9 and Ballistol or Breakfree CLP, but for carbon, they're not ideal. Instead try a water-based type cleaner like MPro 7 or Break Through. They will penetrate the carbon soaking into it and lift it off.
The thing about oily cleaners like #9, Ballistol, Breakfree is they don't need to be cleaned off and dried up and you don't have to wipe the steel with oil afterwards or re-lubricate the action. With the water-based cleaners you will strip the oil off the steel and if there is any lubricant where it is applied it will form an emulsion. They can also damage the finish on wood. That's why people usually avoid them for the worry-free cleaners. But the water-based cleaners are much better at removing carbon. Just remove your cylinder from the revolver and set the revolver aside. Put the cleaner in the chambers with a wet patch and don't soak the ejector rod unless you want to disassemble and re-lubricate it all.
I use a chamber brush as well as a bore brush from Pro-Shot. The Pro-Shot brushes have a brass core and bronze bristles neither of which will scratch steel. I don't use a power tool, but just an all-brass Dewey Rod. This combination works good as long as you're not using a copper solvent like ammonia. For that you'll want aluminum-core nylon brushes, but since your issue is supposed to be carbon, there you go.
