ok well how did your tamagotchi get dirty in the first place? if you are going to a dirty place or somewhere where you might become dirty you should keep your tamas in a bag and never take them out and PAUSE them.
When my Tamagotchi's screen gets dirty or smudgey, I just blow on the screen to fog it up and gently wipe it off with my shirt (you have to do it quickly though; you don't want the fog to get behind the screen ). That usually works for me.