There are a lot of back breaking ways to clean and seal your grout. I have spent hours on my hands and knees scrubbing endless lines of grout trying to get it clean and then keep it clean. There has got to be a better way. This video shows a professional company cleaning and sealing grout. It's an amazing process where the soil and dirt that is caught in the grout is blown away by high pressure jets and then vacuumed up through powerful blowers that ensure the dirty water doesn't get back into the grout.
Most people don't realize that when they mop their floors they are allowing the dirty water to be pushed into the grout by the mop itself. Yes, if you are mopping your floors you're just moving dirt from the tile face to the grout lines. Think of it like water color painting. Every time you dip your brush in the water to rinse out the old paint you make the water dirty. Same with mopping your floor. You should be changing the rinse bucket after every 2 dunks to prevent your mop water from soiling your grout.
Cleaning and sealing grout is difficult and it should be done professionally about every 2 to 3 years. I have been a professional grout cleaner in Tampa FL for several years and I can tell you there is a lot of grout in Tampa homes. This is not an easy task for DIY'ers who like to clean their own homes. Try a professional and expect professional results.