When trying to clean colored grout, you can use several types of cleaners, but try to avoid bleach or ammonia. Both of these substances can discolor the grout. You can purchase a commercial grout cleaning product that should be safe on colored grout.
Effective Cleaners from Household Products
You can also use some substances that you find around the house. For example, you could use a mixture of baking soda and warm water.
Vinegar by itself makes a decent grout cleaner that will not do any damage to the color in the grout, but if you mix it with Baking Soda (Fig.1) and maybe even add salt to the solution, you'll have a powerful cleaning agent ( Fig. 2).
Another effective way to clean your grout is with the use of a steamer. You can also use a portable steamer such as the Bissell Steam Shot (Fig.3) —which delivers great results—or the more bulky commercial model (Fig.4) to effectively clean grout.