How can I prove it is the best interest of the children to remain in my custody after relocating to another state?

There is not a set formula that the court will use to decide custody-- but instead will look at a variety of factors to determine what is in the best interest of the children. 

If you have historically had custody, be prepared to highlight your successes with the children so far.  Note things like their grades, accomplishments in extracurricular activities, and the lack of trouble at school.  If you've done it solo or with limited help from him-- highlight that as well.  For example, who took the kids to school, who has helped them with their homework, and who took them to the soccer games.... if the answer was mostly or always you-- point that out to the court when now he "suddenly" wants to be dad.  You've done the things a true parent does to help their children be well rounded.  Also note to the court that these things were still done even when he skipped his child support payment was late-- you were never late to be the parent.

The second part of your case will be the "going forward."  Be prepared to show the court that you do not intend to give the children any less by moving--in fact, the move is an opportunity for everyone.  Scope out more details about where you will be living and be able to show the judge they will still have access to good schools, programs, and activities.  Of course a judge's concern will also focus on the relationship between the father and the children.  It wouldn't hurt to have an "access plan."  For example, be able to explain how the kids will get to visits with dad.  Arrange/volunteer for video interfacing or skyping with dad on a regular basis to insure that they see and talk to dad as much as they want to.

If you focus on all three parts:  what you have done, what you are going to do, and how you are going to help preserve the parental relationship with dad-- then the judge will see that you are the more responsible parent and you should retain custody.