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Cheapest way is most likely using Luftansa via Frankfurt about $1,700 return although would include an over night stop in Frankfurt the the outward leg

You will get better fares and more possibilities flying from New York JFK to Rome and then get a bus, or train to Naples. Check the Air France site, www.airfrance.us as they regularly have special offers. I fly with them a lot to and from San Francisco and I can recommend them.

Check out hipmunk.com, it's a nice tool for searching for flights. Also consider alternative airports and being willing to take the train from rome (or possibly even milan if the fare is good enough)

Newark (EWR) is your best option for Italy. Also, can I ask, why Naples? It's a one day trip. You might want to review your choices.