You will need a good, coated, one piece cleaning rod. I recommend Boretech rods, they are the best I have seen. If you can not afford to buy a Boretech, save $15.00 or so and buy a Dewey. Cheap, uncoated, and multi-sectional rods damage barrels period.
You will also need a bore guide, the type used from the breach, it replaces your carrier and bolt group. The bore guide will keep your rod centered inside of the bore and will protect the barrels throat area from the jag tip. The guide also keeps solvents out of the upper and will help keep the jag tip centered when you pull the rod back out.
Cleaning rod tips, known as jags, are a matter of preference. With hand lapped match barrels, only the pierce button type or Parker Hale wrap around type should be used. With a common loop type jag, there is the chance of the patch getting bunched up in the bore which can cause one side of the rod to rub hard on the lands, damaging the rod and bore, or the patch can fall off. The loop type jags were made for a back and forth scrubbing motion and it is not a good idea to drag a dirty patch back over a clean section of your bore. When using a pierce button type jag, your patch will fall off after it exits the bore where it can be caught in a bucket or similar container. If doing this in the house, a 3 liter empty plastic soda bottle can be placed over the flash hider and the dirty patch will fall into it. Afterwards place the cap on the bottle and throw it away. This will really help keep mess and solvent odors contained. Every time the rod is removed from the barrel it should be wiped off before reinserting. An abrasive residue will be present on the rod and if it is not wiped off it can damage your bore over time.
A bronze
bore brush can be used and will not damage the bore if used properly. The brush
should be wet with solvent to lubricate it and only be pushed from the breach
to the muzzle, after exiting, the brush should be unscrewed from the rod and
never pulled backwards back through the bore. Use the better quality brushes
that have brass not steel cores. After cleaning, if you blow the brush off with
brake cleaner it will keep the solvent from eating at it and your brushes will
last much longer. Alcohol can also be used. A bronze brush is the only type
that should be used in a match barrels bore. A chamber brush should not be used
in a match chamber and it is not necessary. An exception would be a badly
neglected chamber in a chrome molly barrel. Many match chambers have been
scratched by using a G.I. chamber brush. The G.I. chamber brushes are made for
NATO chambers which are bigger then your match rifles chamber. This condition
also exists with the match M1-A chamber and a G.I. chamber brush.
To clean the barrel, start by cleaning the chamber. I use a .30 cal nylon brush with two .45 cal. patches wrapped around it and the brush screwed onto a pistol cleaning rod or single section from a multi section rod. Wet the patch and insert it with a twisting motion in the chamber, two or three revolutions, and pull it back out. What you see on the patches is 98% percent of what was dirty in the chamber. Do this again with clean wet patches and forget about it for now. What you are doing is letting the solvent work in the chamber. Now insert your bore guide and run a very wet patch down the bore and out the muzzle. Let the solvent work in the bore for a while. Now is a good time to clean the flash suppressor. I use a patch or piece of paper towel to wet the outside, the slots and the inside of the FS. Next would be a good time to clean the bolt face and wipe off the bolt lugs with a wet patch. Set the bolt group aside and turn your attention to the bore. Push a dry patch followed by another wet patch through the bore. Do this 5 or 6 times finishing with a wet patch.
The rifle should be supported by a cleaning rest of some sort with the action preferably locked open with a link device. The barrel should be down hill so the solvent will not run into the upper. Wipe the flash supressor with a dry patch and wet it again with solvent. The goal here is to let the solvent do the work because if you rub on the FS excessively, in time the parkerizing will wear off. Now push another dry patch down the bore followed by a wet one, at this point, the patches should be coming out almost clean. Now is the time to wet the bronze brush with solvent and push it down the bore. After it exits, unscrew the brush and remove the rod. Put the jag back on and push a dry patch followed by a wet patch down the bore, then repeat with the brush. The brush will get the powder residue and copper particles out of the grooves. About 6 cycles of this should be enough. How many patches and brush strokes it takes to clean your barrel will depend on how many rounds have been fired, how smooth the finish in your bore is, and how dirty the powder used, burns. You do not want to overdo it here. Do not over clean the barrel. Dry the inside of your chamber with a clean patch wrapped around your brush. Now take a .30 or .45 cal patch, wet it with solvent, and with hemostats or tweezers, work the patch in a circle around the recessed inside area of the barrel extension just in front of where the bolt lugs engage. Remove the patch, fold it a few times and using the hemostats or tweezers, wipe off the area the bolt lugs engage. Allow the rifle to sit for a few minutes before drying the same way with a clean dry patch. Wipe your FS dry and the barrel is done. If the rifle has a chrome molly barrel, and it will be stored for a while before shooting, a single patch with Breakfree or similar should be run down the bore to leave a light protective coat on it. Be good to your barrel and your barrel will be good to you!