If you are able to disassemble them (...and then put them back together), that's what I would do. You'll need screwdrivers that fit the screws, and a mainspring vise. The hammer is sometimes a bugger to get off the tumbler, and you can probably just leave it on for cleaning. If you want to get the hammer off, there are right ways and wrong ways to go about it, which we can discuss if needed. I would poke holes in a piece of cardboard to retain the screws, and label the screws according to where they go by writing on the cardboard.
For cleaning cruddy old locks, I use kerosene. Put the parts in a pan with enough kerosene to cover and let them soak for a while, maybe just a short time, a couple of hours, or maybe more. Kerosene won't hurt the metal or the finish. I then use a toothbrush to clean off each individual part. For more aggressive cleaning, use a small brass-bristle brush. All of the gunsmith suppliers have these. Wear gloves if you are worried about toxic effects of kerosene. I don't bother, most of the time, but you don't necessarily want to be like me. Wipe each part dry, then wipe it with an oily patch, then reassemble the lock.
If you find you need to clean off rust, bronze wool works pretty well if you don't want to damage the finish or patina.
Good luck with it.
Notchy Bob