I'll see if I can do this from memory. I'm at work and all my notes are at home. Mine has been on the road for 3-1/2 years so I'm not sure if it's still the same process.
First thing you should do is call DMV in Trenton and ask for the group that handles reconstructed vehicles. Tell them you are building a kit car and ask them to mail you an information package and all the forms you need. There are a number of forms that need to be filled out. When the car is almost complete or at the point where it would be driveable, you will need to mail the forms along with copies of receipts, certificate of ownership from FFR, 5 photographs (1 of each side, front, rear & interior), etc. to DMV. After that (about a month in my case) they will contact you to set up an appointment for an inspection. They asked that I have the car on jackstands. They came out to my house and went under and over the car. They seemed mostly concerned with the brake and fuel lines. They suggested a couple minor changes (nothing that I had to do) and then left. They spent about 30 - 45 minutes looking the car over. They said they would send their report to Trenton and I would then get my title in the mail. They were at my house on a Thursday and the following Wednesday my title came in the mail. I then had to go to an inspection station and go thru a regular inspection. My car was titled as a 1999 REC COB. REC for Reconstructed (that's where they put all kit cars) and COB for Cobra. My engine is a 1992 and had to meet 1992 emmision requirements.
Now the good part. Since that time, NJ came out with a "Collector Car" category. Collector cars no longer have to go thru inspection. But, they are limited to 3000 miles per year and they do inspect the odometer every two years. So in any 2 year period you can't exceed 6000 miles. You can go either "collector car" or as a regular every day driver which is what I origianlly did. I have since switched to collector car.
Again, I did this over three years ago, so I have no clue whether it is still done the same way or not. I did not have any problems with any of the inspectors or other people I dealt with. They were all OK. But there are different people depending on where you live, so you very well may have someone different than who I had. The worst part is the waiting. The car is almost done and you're anxious to drive it.
Feel free to e-mail me if you still have questions. Good luck!
Barry