Keeping grout clean can be a bit of a chore. Particularly if you have white tiles, so any discolouration really shows up.
There are plenty of strong chemicals available that promise quick results. Whilst I'm sure they do the job just fine, there is no need to resort to anything that will release strong fumes into your bathroom and harmful chemicals into the waterways.
Scented Vinegar is a fantastically powerful cleaning agent that doesn't use any harmful or toxic chemicals but will blast through the grime and mould on your grouting just as quickly as the more harmful options.
Personally, I use it alongside a Copper Cleaning Pad. They are versatile and flexible enough to get right into your grouting, textured enough to clean it thoroughly but gentle enough to not cause any scratches in the process. Make sure you wet the pad before you use it.