I use a smaller diameter bristle bore brush with a patch to clean the chambers. I wrap about 1/4 of the patch onto the end of the brush, twisted onto the end, and feed the rest (3/4) into the chamber, run it down, twist it around for a while, rinse, and good. I soak the cylinder and barrel in a large size sour cream container full of hot tap water for a few minutes first. They clean up slick as can be with little effort. I just wipe the frame down with a damp rag, then dry, and oil if needed. I don't take the cones out every time, but its a good idea to now and then to ensure they don't rust in.
I usually only dry the chambers, slapping the cylinder on a dry towl or rag to get the bulk of the water out, run some dry patches in, then let it heat dry, and reload the gun. I'll oil the bore a bit sometimes, but often leave it dry after cleaning. I grease cylinder arbor when putting it back together, but otherwise dont do much else after cleaning. I live in a very dry climate.