I use medium bristle toothbrush. If it's a biggen I'd go with a medium size scrub brush. I also use chlorohexadine to do scrubs ounce every other month or so too. Clean and scrub some of the nasties off them for precautionary. Once you scrub you can then apply some mineral oil or vita shell to keep them shiny and good for their keratin too.
As long you don't get in there mouth or eyes. Just a light scrub on the shell and maybe the legs, then rinse it off. Extra care of course while doing it is best. It's definitely not needed as much as I do it. But having so many specimens and species, we have to constantly upkeep outrageous cleanliness and bio security protocol. You would be ok with maybe even dawn soap, just once a or twice a year if you want to do it. Kinda like giving a dog a bath, be cautious around eyes and mouth.
I'm a medium tooth brush guy myself, but they are getting so big it has become harder to use so I switched to one of those lather builders and a wash cloth. They are filthy again within a few days either way. Its funny, the box turtles are never "dirty" like the tortoises.
I wouldn't use chlorhexidine for normal cleaning. For a wound, fine, but it's unnecessary to use a lot of germicidal agents in normal situations. Yvonne, or someone, said Dawn or some other dish washing liquid, that's certainly fine if you need a soapy agent to loosen adherent 'stuff'. Otherwise Tom's approach is very rational.
A clean environment is good and healthy. A germicidally sterile environment isn't possible and is not necessarily healthy for living things. An exception of course for when you have an outbreak of a known disease causing bacterium or virus. When we try to sterilize everything in an essentially natural environment we tend to tilt the balance to the favor of some noxious disease agent or other.