The frequently-repeated advice is to never clean coins and that is absolutely true if you don't know what you're doing - you can irreparably damage a coin by trying to clean it. Certainly never polish a coin. If you think a coin needs cleaning, get some advice before doing anything. If you do know what you're doing, always try on a low-value coin of the same composition first just to make sure. Don't try and rush things either - do it carefully and don't damage the coin in the process.
Avoid chemical cleaning of copper/bronze coins at all costs. Stick with a tooth-pick or something else that won't scratch the coin's surface.
A small amount of white vinegar applied to any difficult to remove dirt or verdigris should loosen it up and eventually remove it. Try and keep the white vinegar to the spot that actually needs cleaning - while cupronickel seems pretty resistent to any major damage from white vinegar, some cupronickel coins are actually a thin layer of cupronickel around a more reactive core (so-called clad coins) and the white vinegar can get through to and expose the core.