The most common way to break in a rifle is to clean the rifle (the bore), and shoot one round. Then clean again. Always clean a new rifle before shooting it - who knows what is in the bore!
Shoot one and clean for at least 10 rounds (preferably 20). This is a very slow and tedious process. It takes me a long afternoon and most of the evening at the range to go through a 20-round break in procedure. Don't get impatient and skimp on cleaning. This is a one-time process intended to give results later when you pull the trigger on game animals. From your avatar, it looks like you are no stranger to long-range shooting.
- Get a cleaning rod guide for the action. The cleaning rod ALWAYS goes in the rear, not down the muzzle.
- Use a good one-piece cleaning rod (Dewey rods are good). Do not use the crappy screw-together cleaning rods (unless it is an emergency of some kind).
- Use nothing harder than a bronze bore brush. Stainless brushes are out of the question.
I don't consider the one-piece rod and rod guide as optional. More rifles have had their accuracy ruined by poor cleaning practices than any other cause.
The purpose of the "shoot one, clean one" method is to have the bullets going down a clean bore so they can polish it. They will leave copper fouling as they do this, so you want to clean it out before the next bullet passes. This allows the bullets to take out small machining burrs or other imperfections without copper build-up.
Soon, the bore will be much easier to clean and will not foul as soon as it would without the barrel break-in. Accuracy will also become more consistent. You'll get fewer flyers.
For cleaning fluid I use Shooter's Choice or Butch's Bore Shine. Either of these will do a good job of removing copper & carbon fouling without the aggressive tactics of most dedicated copper removers (heavy ammonia). My procedure is to run two wet patches down the bore, then carefully run the bronze brush all the way back & forth down the bore a few times (never reverse halfway!). Run one more wet patch and let it soak for 5 minutes. After the wait, run another wet patch. This one should come out with some green showing. If you're seeing black, run another wet patch and use the brush again. The black is carbon/powder fouling, though after only one shot the first use of hte brush typically takes care of this. The 300 RUM may have more than most, though.
Keep it up with the wet patches until the green goes away. You'll see more after brushing - the bronze brush cause a bit of green to show by itself.
After the patches come out without any green (or very little), run a couple dry patches down the bore and shoot again (do not shoot with a wet bore).
After several shots you will probably notice it takes less time to clean the bore. The bore is getting smoother and less copper is getting worn off the bullets as they are shot.. which is the whole idea!