I like to clean over a towel. I usually lay the open revolver on its side and spray some CLP in the barrel and chambers then I turn the gun over to coat everything. Then open the cylinder and do about 10 scrubbing passes with a bronze brush on each chamber, I usually loose count or do a chamber twice so it normally works out to more. Then I scrub the barrel the same way, back and forth, and take a brush to the forcing cone and cylinder face. Then the extractor end of the cylinder and the recoil shield on the frame. This gives the chambers and barrel time to soak, then I scrub them again, wipe everything down, get a patch and do the chambers. I do this a few times 'till the patch is pretty clean. next I run a patch through the barrel followed by a few more till they're pretty clean too.
Then I check under the extractor which can get way over lubed and under cleaned, and square that away.
Then I'm pretty much done, 10-15 minutes in all.
For a more thorough cleaning I'll remove the sideplate, yoke and cylinder. More cleaning, the lockwork comes out too. Each piece is cleaned checked for wear and any scratches from rubbing the frame, I usually stone the rebound slide, re-lube and re-assemble.
For leading I like the lee lead remover and a good soak with Birchwood Casey bore solvent. That combo always does the trick. In general I like CLP since the one product can clean and lube. I have found that some of the new water based solvents are great on stubborn carbon fouling but lubing everything after using them is almost the same work as using CLP from the get go.
The only thing you really have to be careful about are Stainless Steel bore or chamber brushes, re-lubing after a water based solvent, and Hoppes on nickle. I've used SS brushes on REALLY bad leading, but working very carefully. Soaking and a lee lead remover is less risky and really works well for me.
/c