I clean my breech plug by first wiping it manually on the outside with a patch soaked in #13. Then I use a bore brush to scrub all the threads. Then I take the wet patch, shove it into the concave end and turn it with a small flathead screwdriver until the inside is shiny clean. I then pour some #13 into the plug and let it dribble out the other end. I clean the primer end the same way, with a small flathead screwdriver turning a wet patch. If a visual inspection reveals any obstruction in the hole, I pick at it with a sewing needle and drip more #13 through it.

That's for field cleaning. At home I just soak the entire thing in #13 for a while and then rinse off with hot water.