First of all - do you have your NJ CCW? If not, I would say just leave it at home...
If you do, then do what I do when I travel out of NJ.
I'm assuming it's a handgun, not a rifle/shotgun...
You will have to pay to check a bag. In the bag you should have a locked, hard-sided case (I use the same lockbox I do for my car). Put the gun inside the case, unloaded, mags inside the same case. Ammo, I bring a box of Hornady Critical Defense - in the original cardboard box. TSA states that small arms ammunition “must be packaged in a fiber (such as cardboard), wood, plastic, or metal box specifically designed to carry ammunition.” I usually put the ammo in the same case, however I've had airline check-in's once tell me to just take the ammo out of the locked container and leave it in the luggage. YMMV.
When you get to the airport, go to the baggage check in. Go to the counter and tell the agent that you're declaring a firearm. You’ll have to read and sign a tag saying you're complying with all the firearm regulations and then insert that tag into your checked luggage. The TSA inspection will take place at the check-in counter or may take place at another TSA area (they will escort or point you to the area). Make sure you KEEP your luggage claim tag.
They should not ask for the key/combo for the lock. DO NOT USE A TSA-APPROVED LOCK. If they need to open the case for inspection, they will tell you or page you. Make sure you check in with enough time to deal with the counter and with the additional security.
When you arrive in NJ, they may tag the luggage as requiring special handling since there's a firearm inside. If you don't see your bag at the baggage carousel, go to the Baggage Claim office for the airline and tell them you had luggage with special handling. They'll check your ID and luggage claim tag before releasing it to you.
Hope this helps.