My dog has a fountain waterer which has a charcoal filter. it’s very simply constructed: a fiber-mesh bag with a handful of charcoal pellets loose within it. It gets gunky-- a brown algea (at least that’s what I think it is) coats it after a while.
I’ve been cleaning it by rinsing it in the sink until it looks clean, but that’s not exactly cleaning it, but I know any soaps or cleansers would be absobred and would come back out into the drinking water.
Well, a bleach solution will kill any bacteria or algae growing on it, but after a while, the charcoal will have absorbed so many contaminants that it is no longer an effective filter, except for gross particulate matter. At this point, it needs to be replaced. In theory, you can reactivate the charcoal by heating it, driving off the adsorbed contaminants, but this takes quite a bit higher temperature than your typical oven can deliver - between 800 and 1000 degrees F, IIRC.
Oh, and to answer the question you are going to ask, yes, bleach is safe for this purpose - once it fully dries, it leaves only sodium chloride (salt) as a residue. It’s used in food prep areas as a sanitizer for this very reason.
The typical solution strength for sanitizing is about 1/2 ounce to an ounce of bleach per gallon of water. You can use a stronger solution, if need be, but at higher concentrations, you run the risk of causing oxidation damage to the item you are trying to clean, particularly if it has plastic parts.