I fill the barrel with dishwater. Nothing but water and some Dawn Dish soap. I am so cheap I even take it from the sink, after doing the dishes.
I have a small funnel and an empty dish soap detergent bottle (Ajax) that is filled with dish water. I put the tooth pick in and hold it in place with the frizzen. Then insert the funnel and fill the barrel. Set that to the side and get all the other stuff put away.
I bet it only took five patches to have a clean patch come out. Then I alcohol swab the bore and then dry patch it. After that I use Birchwood Casey Barricade to protect the bore.