Start by adding baking soda to a bowl. The amount will depend on how much grout you have to clean. For a small entry, start with about 1/2 cup. For a larger bathroom floor, try a full cup. (Remember, you can always mix up more if needed)
Add hydrogen peroxide to the baking soda and stir. I recommend adding a few squirts of hydrogen peroxide, then stir. Add a bit more. Stir. Repeat this until you get a consistency that resembles soft toothpaste. You want it to stay on the brush and not drip off. If you ever feel the consistency isn't right, you can always add more baking soda if it's runny or more hydrogen peroxide if it's too dry.
To use, scoop up the mixture with a toothbrush (or grout cleaning brush) and scrub onto the grout lines.