Starting Out
If your coins are so heavily soiled that you cannot even tell exactly what they are, then the best way to move forward may be to give them a "bath" in mild dish detergent and warm water. You can use a child's toothbrush to gently scrub grime off the coins, but do not use hard, sawing motions to clean the coins or you could end up eroding away brittle metal as well as dirt. However, this type of cleaning is only appropriate for coins that are still in circulation. If you are dealing with proof coins or uncirculated coins, then they should never be immersed for any reason. Instead, immerse your fingers in the soapy bath and then rub the coins gently between your thumb and forefinger. Wipe the coins clean with a soft towel but do not rinse them under running water.
Avoiding Scratches
Surprisingly, coins are easy to scratch even though they are made of metal. Just banging them against each other in your pocket can lead to some serious damage, so do not pile them up as you clean them. Always dry coins individually on a clean cloth and then lay them on a clean, dry towel to dry them completely. If you feel any grit on a coin while you are rubbing it, stop rubbing immediately so you do not create scratches on the surface of the coin.
Dealing with PVC Residue
PVC residue is generally a result of storing coins in plastic coin sleeves. It is light green and tends to smell acrid. You can safely remove PVC residue from your coins by dipping a Q-Tip in acetone (nail polish remover) and rubbing the Q-Tip gently and repeatedly across the surface of the coin. When the Q-Tip becomes covered with greenish sludge, move on to a clean Q-Tip so you do not just smear the residue around.