What do I have to do to move out-of-state?

While generally one ex-spouse cannot stop the other from moving to another state, especially if there are legitimate reasons such as relocation for a job, it's important that the other parent has the ability to visit with the children. To make sure that there are no misunderstandings, you should bring up the fact of your desire to move to New Jersey to the court. Since your ex is not agreeing to it, you need to get the court to grant your request and incorporate it into your divorce/custody agreement. Then everything will be legal. But be sure to show the court that you are willing to do whatever is necessary for your ex to see and visit his children in New Jersey or to have the children visit him in Illinois.