There are two kind of cleaning for clay teapot. One cleaning refers when the pots are brand new. The other cleaning refers to maintenance cleaning.
There isn’t much disagreement for how to clean a brand new pot. Boil the entire teapot in pure water or water with a bit vinegar or water with a baking soda (not harsh chemical) and also brush with a very soft brush. Clean the brand new pot until you do not see any dust or fine particle. Usually take at least two cleaning cycles.
There are two line of thoughts for regular cleaning. The most traditional and most widely accepted thinking is that you do not need any cleaning. At most, rinse the teapot with boil water a few times. The idea is to prevent damage to the pot and also keep the tea reside fragrance in the teapot. You can think of it as seasoning the pot.
The other approach is almost the same as the above, except allowing occasionally cleaning with acidic water like vinegar with water and alkaline water like baking soda with water to remove excessive the tea residue or funny taste, and then rinse a few times with pure hot water to remove the vinegar or baking soda taste. This vinegar water and baking soda with water cleaning is only to be done as needed, maybe like once every 6 month or once every few years. This is not to be done routinely.