Here is a tutorial for that https://www.hometalk.com/640811/the-absolute-best-way-to-clean-grout-4-methods-tested-1-clear-winner
I stick to bleach, still works best but does smell.
In our old apartment, I used gel toilet bowl cleaner and a toothbrush to clean the grout. Here's a tutorial I found for this method:
I use Clorox Toilet Bowl cleaner which does a great job for me. I have tried many other suggestions but nothing worked as well as this one. It is a gel. Place a dab in the corner of four tiles, spread to the grout lines with a toothbrush. let sit for 10 minutes and then use the toothbrush again to scrub a little. Wipe the area up with a sponge dipped in warm water and vinegar to get rid of the soapy reside. Then steam mop or damp mop the whole floor with a water and vinegar solution again. Do not get the Clorox on your wood floors.
This is the shower cleaner that we use to help keep grout and tiles clean and minimize the mildew - it works on kitchen grout too. Hope this helps! Hugs, Holly
PS - I'm getting ready to release a FREE eBook with recipes for all of my homemade and all-natural cleaners. If you want to know when it's released, you can get it here --> https://pinkfortitude.com/thank/