If it's just grime... acetone will clean the silver coin.... but what you may think of as grime may actually be toning or tarnish of the silver...
Acetone will not clean off tarnish or toning... you either need some sort of acid to clean the coin (DO NOT DO THIS IT RUINS THE COIN'S SURFACE 99 OUT OF 100 TIMES!)
Or you can reverse the chemical process.... if you're interested in doing that.... I recommend you talk to your science teacher at your school... do some research and try to do this for some extra credit...
Personally.. I wouldn't bother... let the coin stand as is... if it's not clean to your satisfaction after using acetone than look to trade or exchange that coin for one that's more eye appealing.