When you say "natural tile", I am assuming that you are not talking about a glazed porcelain or ceramic tile. In this case, you will need to clean the whole tile, including the grout lines. You can find a quality cleaner in the home improvement stores. In this case, the adage "you get what you pay for" fits. Some of the tile cleaners are expensive, but the more expensive usually the better they work. Once the tile is cleaned to your satisfaction, it needs to dry completely before sealing. Then once the tile and grout is ready for seal, purchase a high end sealer and seal the ENTIRE surface. Depending on usage of this area, you should reclean and reseal this tile and grout every 12-18 months.
Many natural stone surfaces are very porous. If you have a marble or a granite it's possible that you have stained the tile itself in which case no amount of cleaning will bring in back new. Find out what type of material you are dealing with and go to a specific stone and tile distributor with pictures they will be able to assist you best.
The type of material that the tile is makes the difference. You will need to find a product made to clean the different types of stone, I never recommend anything like Tilex since it will etch some stone and grout. After you get the stone and grout clean you will need to seal it, I'm not sure what products are available in your area so I cannot tell you what will work best.
Hydrogen peroxide is a good cleaner for natural stone and grout. It is pretty pH neutral.
Check out this article I wrote