Please see my thread on the 10/22 Hobby. I have an unusual technique on my 10/22s. First...I always clean from the breach to the muzzle. Since I have learned how to make the 10//22 a precision weapon, I shoot a lot. Accurate rifles are fun....this leaves a lot of crud inside.....so....
I remove the stock and field strip the barrelled action. I leave the barrel on. I then point the muzzle into a trash can and spray the inside of the receiver with aerosol brake cleaner from the local auto store. 1.99 on sale. This works the crud loose. I then use a tooth brush, pipecleaners, and a shaving brush to remove the junk that the brake cleaner didn't get.
I have modified my 10/22s so that they can be cleaned with a cleaning rod from the breach end (please see my 10/22 thread for this). I like stainless steel rods or bore snakes. Both work well....sometimes a brush is needed so I use the bore snake for quick jobs and the cleaning rod for deliberate work.
I use whatever bore cleaner I got on sale and follow it with a light coat of oil. My 10/22s all have lapped barrels, again the thread, and are pretty easy to clean.
I use DuPont Teflon lube for the bolt and trigger assembly, a small dab of Riggs on the receiver rail and put the whole thing back together...It takes longer to talk about it than it does to do it.