You never want to clean out the slime. It's usually full of benificial bacteria and other stuff that other fish can feed on.
Many people have different definitions for full tank cleaning. You never want to take out the fish. People normally remove from 20% to 30% of the water and replacing it. When you replace the water, you should add dechlorinator, which would remove the chlorines/chloramines from your water supply. Prime is commonly used, becuase it also helps to bind with the ammonia (chloramine - chlorine = ammonia).
You can remove more than 30% IF and only IF the water comes from a container that has been stored for a few days, with dechlorinater and also heated. This way, there is not a big change in water parameters from the old and new water.
When performing a gravel cleaning, NEVER change the media in the filter the same day. This may disturb/remove too much benificial bacteria from your tank and cause a mini-cycle.