One way to clean a PS2 disc is to breath on the back of the disc and gently wipe off the back of the disc with the inside of a T-shirt. Another way is to put a small bit of toothpaste on the back and wipe that around it helps fill in the scratchs.
Using a lint free cloth, wipe from the center of thd disc directly to the outside edge. Do not press hard, use gentle wipes. You can use alcohol to remove dirt or fingerprints.
Use window cleaner and a microfiber cloth. Don't use paper towels or an ordinary terry cloth towel because they are abrasive and will scratch your discs.
Most of the time I always try to keep the case available for storing my games so they don't need to be clean. But if for some reason that is not possible than I know one good investment is a microfiber cloth. Now I personally do not use any spray because I fear build up.
You should clean a PS2 disc with a soft linen cloth lightly dampened with alcohol, wiping from the central whole out toward the edges. Then let air dry.