What I am going to suggest produces lead oxide byproducts which can be rather lethal, so using rubber gloves and paying careful attention to cleaning up any materials used is ESSENTIAL. I would also advise you ignore any stray itches while you are wearing those gloves.
I'll also warn you that it's not a fast process, you can plan on 8 to 16 total hours of work to get that barrel fairly clean.
Items needed, at least 38 caliber bronze bore brushes, some pure copper Chore Boy (test it with a magnet because there are a lot of copper plated steel imitations out there), a cleaning rod, 38 caliber Jag, patches, a bore mop, and finally a bottle of Peroxide and White Vinegar.
Mix the White Vinegar and Peroxide 50/50 in a disposable containter deep enough to dip a bore brush into. Start with mopping the barrel with your cleaning mix with a soaked bore mop. Let it soak for 15 minutes to a half hours and if it shows signs of drying mop it again. Next, take that Chore Boy and unwind a bit of the copper wool. You want to put ONE wrap of the Chore Boy around a freash bore brush and force it all the way through the barrel. Take it all the way into the frame window and then pull it out of the barrel. Do this 3 to 5 times and then dispoze of that piece of Chore Boy. Mop the barrel again and let it soak some more. What you are doing this soaking period is allowing that peroxide vinegar mix convert the freshly exposed lead to lead oxide which is much easier to scrub off. Keep alternating the Chore Boy/soak process until you see that the lands and grooves are mostly clean. BTW, a LED penlight poked into the chamber end of the barrel makes it easy to see your progress, Streamlight has one called the Stylus that I carry everywhere. Once the lands and grooves are mostly clean, it's time to get the nooks and crannies using a bare bore brush in the same alternating process, however it's helpful to wet the bore brush with your mix. Finally, except the fact that the barrel doesn't have to be 100% perfectly clean, a tiny streak of lead in the recess next to the lands won't hurt anything. Once it's good enough, patch the barrel dry and then oil it well using a series of oil soaked patches until they show clean. Note, this progression of oil soaked patches should act to remove any traces of remaining vinegar peroxide, so a more is better approach is quite sensible. BTW, I like Mobil 1 synthetic because I've found it great for keeping a freshly bored engine block from rusting in the bores. IMO anything that keeps freshly machined cast iron from rusting will also keep gun barrels from rusting.
Naturally, it's a rather wet and sloppy process so you'll want to use a good layer of newspaper to soak up any drippings and dispose of them properly. You'll also want to either protect blued areas of the frame and the lockwork from any drippings as you proceed, that mixture can be mildly corrosive. Keep a roll of paper towels handy and dry off any areas that aren't leaded ASAP and drop the contaminated debris in your disposal bag. I'd also suggest that you plan on removing the lockwork for a good cleanup and re-lubrication after this process.