I have found the best and easiest browining solution is made by Mountain Laurel Forge. It is a cold browning solution. Do not polish your parts beyond about 220 grit emery paper. The browning solution needs a little bit of texture in the metal to work. Something to grab onto. Thoroughly remove all signs of grease or finger prints with acetone. Wipe on a coat of the browning solution and place your barrel in a humid place. I use a piece of 4 inch PVC pipe with a plug in one end. I put a cup of water in the botom and hang the barrel in it. Let it stand for about 24 hours and then remove it and rub it with a rough cloth. A piece of burlap works best. Smooth out the surface rust the best you can. De-grease it again and recoat it with the solution. This time let it set only a couple of hours. The time may vary. When you see another coating of rust, again rub it down with the burlap. Degrease it and re-coat it with the solution. These next few coats will require close watching because as the browning process gets started, it speeds up. Do not let it get away from you or you may get some surface pitting. Once the barrel gets to the degree of brown that you want, just remove it and wash it with a baking soda solution to stop the rusting process. At this point, I thoroughly rub it down with a good gun oil such as Ballistol. If you have it, a good coat of bees wax that is rubbed in and buffed thoroughly is traditional. If you don't have bees wax, you can use any paste floor wax. I like to use a wax called Renaissance Wax. It is available from many gun dealers or fine furniture stores.
The process is not at all difficult but it does require that you keep an eye on it and not let the browning process get away from you once it starts. Laurel Forge makes a nice soft flat traditional brown. If you want a more shiney brown, you can use Plum Brown by Birchwood Casey. If you use it, you can polish the barrel a bit more to get the shiney brown finish. You need something like a propane torch to heat your barrel with to get the browning solution to work. You get the barrel hot enough so that water will sizzle on it. Then you wipe on a coat of the browning solution and let the barrel cool. Polish it with a piece of rough cloth and repeat the process until it is as dark as you like. I have used both and the Laurel Forge cold browning is easier to use but takes longer. For small parts, you can't beat Plum Brown. Just hold them with a pair of pliers over the kitchen stove until they are hot enough and swab on the Plum Brown. Two or three coats and you have a beautifuly browned part. It is the larger parts that are just a bit harder to get evenly heated and coated for an even brown. Oh, you will eventually get there. No doubt about that but it is just harder to heat and coat a large piece like a barrel. At least that is my opinion based upon having browned several rifles and using both Laurel Forge cold browning solution and Plum Brown. Either will get you there but neither is an instant browning solution. Have patience and work carefully and follow the instructions that come with both.