Outofstater, I don't know if it's the "Right" way or not but I have good proven results cleaning my BR guns this way and I've looked though the barrel more times than not with a Bore Scope through the process. A good cleaning rod and a bore guide are a must, I have a couple Dewey rods and a Tipton, I like the Tipton better, there are a lot of good rods out there.
I first run 2 patches of solvent, Use you choice here, I don't want to start an argument. Then soak a bronze brush with solvent and run it back and forth 10 times, sometimes I'll do 5 passes and put more solvent on it, once you get a feel for it you can almost tell when you need to do it. Then I run solvent patches though until they come out clean, pass dry patches though until they are dry. This will get rid of most of the carbon fouling, maybe not all.
Now onto copper, the best stuff I've found is KG-12, it is wonderful stuff and the most aggresive on copper, once again verified with a bore scope. Follow the directions and it will work wonders, soak a patch and run it back and forth 20 times, don't use a bronze brush, the KG-12 will eat at the brush. KG-12 isn't ammonia based, your patches will not come out blue, the bad part about that is without a bore scope it's very hard to tell if you have all the copper out. Once I'm done with that I run dry patches though until they come out dry.
Back to 2 patches soaked in solvent, then soak the brush and run it through 5 to 10 times. Run solvent soaked patches though until they come out clean, then run dry until they are dry. I like to finish up with a patch with a few drops of Kroil to protect against rust.
There are several good carbon solvents on the market, I never really compared them to see which cleans better. If things get real bad you can use JB bore paste or Flitz, I usually use one of those once throughout the season and then again at the end. But keep in mind, these barrels are getting between 800 and 1200 rounds through them a year, A hunting rifle won't see that many rounds in it's lifetime, well most wont anyhow.
My deer rifle is a straight factory 25-06 Remington, that barrel is so rough inside, I do that cleaning regimen at the end of season and call it good, I know there is still copper in it but so what, a couple shots a year and it's still accurate enough to shoot a deer at the distance I expect.