The best way to clean grout is with a quality dry vapor steam cleaner.
The steam will penetrate the porous surface and float the soap
bacteria and scum out so you can wipe it up. If you have the common
two inch tiles with eighth-inch grout lines, you could use a brass
brush to lightly abrade the surface if the nylon brush doesn't give
desired result. As a last suggestion, use one part phosphoric acid to
four parts water. This is an etching compound and will erode the grout
surface after short exposure. Once the acid starts bubbling the grout
line, rinse thoroughly down the drain. It is important to remove all
residue because the chemical will even etch the ceramic on tile.
Unfortunately, nothing but grout replacement may work at this point.
The
bleach has likely changed your grout color.