Here are the basic supplies I use to clean the bore of my rifles along with a shooting rest to hold my rifle while I clean the bore.
I have read many a comment and article on how frequent and what process different shooters use to clean their rifle barrels. It seems that everyone has their own opinion on how and when they clean their rifle's bore.
I have in the past done the tedious process of cleaning the barrel after every shot, over and over to break-in a new rifle... And then I have done very little cleaning to a new rifle barrel.
What I have found is that if I give the rifle a good cleaning or two removing all the copper fouling during the first twenty or so shots that it is plenty for breaking-in my new rifle barrels. A new rifle will have already been test-fired at the factory so I clean the rifle's bore removing all copper fouling before I fire a single round and one or two more cleanings within the first twenty rounds. Then I often shoot hundreds of rounds between cleanings and often clean only because its the offseason and am looking for a reason to open the safe and spend time with my rifles.
Now, if I ever switch from a molly coated to an uncoated bullet or vice versa I always clean and remove all the copper fouling. And if a rifle isn't likely to be shot in a long while I will run a patch soaked in Kroil through the bore.
I do feel that cleaning can actually do damage to the riflings if not done properly with the right chemicals and equipment. I really like Bore Tech's Eliminator bore cleaner as it does a great job at removing copper fouling and I don't have to worry about ammonia and the possibility that it could damage the bore. After using the bore cleaner I run a couple of dry patches through the barrel and then a patch with Kroil on it and I'm finished.
I also use a good cleaning rod and rod guides designed for my individual rifles. I use correctly sized Dewey cleaning rods and Sinclair International's cleaning-rod guides. The correctly sized rod guide keeps the cleaning rod lined right up with the bore. Not using a rod guide or an incorrectly sized guide will likely lap and damage the throat of my barrel, in turn throwing accuracy right out the door.
I have really found no accuracy difference in how a rifle shoots based on how often I clean and removed the copper fouling. Yes, I believe barrels can lose accuracy if they get too much copper fouling, however, as infrequently that I clean my rifles it's still frequent enough to keep them more than accurate. My rifles all shoot just as good with ten rounds or 200 rounds having been fired between cleanings of the bore.
The most important thing I have found is that I really enjoy spending my time shooting rifles, not spending my time cleaning them. I have also found that the first shot out of a clean bore is often going to be a little off course. I find that after a barrel is cleaned it needs to get fouled up properly to get consistent accuracy. After cleaning the bore I always consider that my first shot is a fouling shot before I start looking for tight bullet groups.
Update: April 2017
I have adding another step to my cleaning process to polish out a rough bore and increase accuracy.