I've never had much luck cleaning electrical connections that were very corroded (spelled green crust) with commercial cleaners, although there are probably some that work.
Soaking the connector in vinegar for a while helps to loosen up the crud. Then, using a non-metallic tool, scrape off what didn't dissolve or fall off. Once the really bad stuff is gone, you could try the commercial cleaners for your finishing touches. Don't forget to use dielectric grease this time. This will keep the corrosion from coming back.
If you use a metal object to scrape the connections, you will clean it quickly, but you will also scrape off the plating on the connection. This will decrease it's efficiency, and leave it more vulnerable to future corrosion.