If it is a bolt action then clean it from the receiver end and use a bore guide if you really want to be careful.

Run a soaked patch with your solvent down the barrel and let it sit for a couple of minutes.

Run your brush down the barrel a few times.

Run some dry patches down the barrel to get the gunk out.

Repeat if necessary

After you get all the fowling out then run a lightly oiled patch down the barrel.

Run a dry patch or two to remove the oil before you shoot it again.