Transferring on an international flight
 

We are traveling to Athens, Greece from CLT. We transfer at JFK for flight to Rome. Will we have to claim our luggage in Rome, go through customs and re-check prior to boarding our flight to Athens?

You would do better to post this on the Europe or Air Travel boards . . . but, assuming they are on one booking (or on airlines w/ interlining agreements) you will check your bags through, you will go through immigration in Rome, and not see your bags or go through Customs til Athens.

Is it all on one ticket, all on the same airline?

It is all on one ticket...Delta with a codeshare airline.

All on one ticket as Janis said. You go through immigration (passport control) in Rome, and probably through security again. You collect your bags in Athens and go through the green, nothing to declare lane. It is highly unlikely that you will be stopped by a customs officer.