Using 1/4 cup of vinegar, mixed with a couple teaspoons of salt, should be enough to clean any handful of coins. The mixture shouldn't be diluted with water. However, the coins should be thoroughly rinsed after soaking in the salt/vinegar solution.
Metals commonly react with oxygen in the air to form oxides, for instance iron oxide, commonly known as rust. This same thing is true for coins - an oxide will form on its outer surface over time, giving it a tarnished appearance. By submerging the coins in a solution with salt and an acid (vinegar), the oxide can be removed.