Using bleach can be a smelly process, but it is one of the best ways of killing mould and its spores. I'd suggest after a thorough cleaning for you to seal the grout with Davco 500ml Sanitized Tile And Grout Sealer. The mould is hard to remove because it grows deep into the pores on the cement-based grout, and no matter how much you scrub, it won't come out easily. The sealer fills these pores so the mould can't enter them.
You should also find this step-by-step guide useful: How to seal grout.
Wear a respirator to prevent inhalation of fumes, but the key is to have the room well ventilated, so open some windows and put a fan on in there to circulate air.
Some materials you'll need:
- MitchellMc