This is what I do when cleaning my rifles:

1) Bore brush, pass through 2-3 times. (Maybe more if really dirty)
2) Pass through a couple dry patchs
3) Attach the bore brush and place a patch with Bore Solvent (Copper remover, etc...I use Shooter Choice Copper Remover) Pass it through several times. If coated I will replace and pass a clean patch with another application of solvent. I would say after a number of firings I will run it through 10 times.
4) Run dry patchs through with bore brush to remove solvent, until clean
5) Pass a patch with remoil through once
6) Pass Dry patches through until clean
7) Thin coat of Remoil on stock, barrel, etc
8) Lubricant (Anybody try the drilube by Rem?)

Make sure to place scope cover on lens, use a soft cloth to remove excess Oil on the barrel, stock, etc.