J-B Bore Cleaning Compound first
Before you fire the first round, thoroughly clean the bore with J-B's. Assuming its a .308, use a 7mm bore brush (one size smaller than .308) to hold the patch and then clean and polish the bore as the directions indicate. Make 10-15 passes though the bore with each patch and repeat for 3-5 patches. Do not let the brush and patch exit the bore at the muzzle as a small amount of damage/rounding to the crown may occur (or so I've been told). I mark the rod so I know how far to push the rod. It's slow and tedious but worth the effort. J-B's will clean out all the fine pieces of metal remaining from cutting the rifling and will also polish the bore to some extent. After the J-B's, swab the bore with Kroil, then dry patch before you go to the range.
As far as cleaning between shots, the idea is to shoot the first few (10-20) bullet's through a clean, dry bore. I clean after each round for the first ten, then at 15 and 20. Then it goes home for a thorough cleaning with a heavy ammonia solvent (eg., Sweet's 7.62) and maybe another round with J-B's. After that I wouldn't worry about break-in.