The best moving company in northern NJ that I've used is Casella and Sons, hands down!
I've used them myself, and a few of my friends, family members, and co-workers have used them multiple times as well. I can't say enough good things about them. And no, I don't work for them, and I'm not related to them.
They're based out of Verona, but they cover all of NJ.
Casella and Sons
973-239-3278
Casella & Sons Moving Service