I have found that Laurel Mountain Forge browning solution to be the easiest to work with. It is a rust browning solution so it is much slower than Birchwood Casey's Plum Brown but it requires no heat like the Plum Brown does. The Laurel Mountian Forge directions
http://www.laurelmountainforge.com/barrel_brown_inst.htm are pretty clear as to how to get a nice brown but from personal experience, I can tell you that the first coat or two may take overnight to start leaving a nice coat of rust. It is important not to over polish your barrel before browning it. The LMF browning solution needs a less than brightly polished surface so it can get a start. If your barrel is polished too much, there is nothing for the LMF to work on and it will take much longer to start working and may not give you an nice even brown. A barrel with a satin finish will work best. Once you get a rust coat started on your barrel, the next coats will take much less time and you need to watch it so you don't get pitting on your barrel instead of a nice smooth rust coat. I found that after you get a rust coating started, the following coats will need to be watched almost every couple of hours so you can remove it from the humidity chamber and card off the rust before starting another coat. Keep applying coats of LMF, letting it work and then carding it off and washing the barrel until you have the depth of brown that you want. it is not hard, you just have to keep an eye on it so it doesn't get away from you.
I use a piece of 4 inch PVC pipe with a cap on each end as a humidity chamber. Glue the cap on one end and put a few inches of water in it. stand it up absolutely straight and hang your barrel down in the pipe, Put the cap on the other end to keep the humidity in. Put the pipe in a place that will keep it warm. Be sure that barrel hangs free inside the pipe and doesn't touch the sides. The first coat will take about 12 hours. The second coat may take just as long but by then , you should have a good coat of rust started. Each coat after that, will take no more than 8 hours. it usually takes about 5 to 10 coats to get the brown that you want but it is a very durable brown and I really like it better than the Plum Brown. It is more authentic, I think.
Buy a bottle and try it, What's the worst that can happen? If it doesn't look right, just use emery paper and take it off and start again. Even if you have to start over a couple of times, you will finally end up with a beautiful barrel if you stick with it. It's easy, you just have to watch it and not let it get away from you.
Good luck and post a picture of the finished product.