How do I reach NYC's East village from JFK Airport using public transportation?

Yes. But its not easy. You can take the AirTrain to Jamaica Station, then the LIRR into Penn Station (or better yet, Flatbush Avenue). From there you can connect to the subway.

You can check out this site: MTA Home Page (http://www.mta.info/) for specifics.

Where exactly are you trying to get to.