It doesn't get much more elegant than this steam cleaner from Pedersoli, but Harbor Freight has a much cheaper alternative that can be adapted. My preference when using a rifle vise for cleaning is to place the stock up (unlike the video) and the muzzle down so liquids are less likely to leak into the channel under the barrel. Since you have to clean a trapdoor from the muzzle, a bore guide is useful to prevent damage to the crown. Plain hot water works fine. Wet patch, dry patch until clean.
Make sure you have container of soapy water to put the fired brass in until you can wash it out prperly later when you get home.