I usually use a small patch or cut rag and one of those hooks that comes with cleaning kits to get in there. 

Was cleaning my new light weight build after a short range session/ sight in and was thinking there must be a better way. 

My tip for the day...

One thing I do use to clean the chamber is attach a 30 cal. brush onto a section of cleaning rod and insert that into a low rpm drill/screw driver. I ten take a rag soaked in solvent and wrap that around the brush. Insert that into the chamber and spin the drill. I can get the chamber shining like a mirror with a couple passes.  

I sometimes put a dab of Mothers mag cleaner on the rag. That seems to really help get rid of and carbon in the chamber.

Again, Im using very low RPMs as to not damage the chamber. 

Spritz of cleaner, rag and a q tip get the job done. No special ritual or compound. As for the chamber: get a chamber brush, simple solution to simple problem.

Then again I do not clean my rifles as meticulously as I used to save for once every so often.

I push it up, qtip with some clp.

Chamber gets a chamber brush if I'm going to do that. Usually just a qtip with clp swirled around to get anything that may have fallen out of the barrel extension when I swabbed that. Finish the area with a bore snake.

Usually takes about 15 minutes for a functional cleaning.