The correct way would be to let the power down with the correct size watch key, by moving the click away from the ratchet, that is located under the little plate, held by the tiny screws. Always remember with a cylinder escapement there are no pallets, so power must be released first before any attempt is made removing the balance.
So you have the barrel and the bridge all together. Remove the barrel cape and remove the spring. The barrel arbor should unscrew and become two pieces, and then it will come away from the bridge. Many are not easy to take completely apart, if you find this my advice would be to clean it as it is, you do not want to damage the arbor. You have the thin plate off which exposes the ratchet, that is the part that is going to be the most dirty part and underneath.
I cannot remember which way the arbor unscrews. Its over 25 years ago for me. Not all arbors have the two holes as in the diagram above. so be careful.