Actually, adverts for cleaning products often describe how the product "dissolves" dirt, grease, grime etc. so that you can easily wipe it away. By way of example, see the above search for the phrase "quickly dissolves dirt".

Edit: Alternatively, you can also say that your product removes dirt, grease, grime or whatever.

dissolve is really the correct word, with remove being a close second.

But some other options:

- gets rid of grease and stubborn (or tough) stains,
- fights grease and stubborn (or tough) stains,
- cuts through grease and stubborn (or tough) stains,
- is tough on grease and stubborn (or tough) stains,
- breaks down (as in decompose) grease and stubborn (or tough) stains, etc.