Are you married and do you have a college degree?
The Army restricts the number of dependents and single parents because it understands that you have responsibilities as a parent and--if it comes in conflict with your duties as a soldier--the Army tends to lose a soldier. If you aren't married, you're essentially a non-starter. If you are married, the likelihood of getting a waiver goes up with your ability to support them better. This means entering at a higher rank (i.e more money per month), the ability for your spouse to contribute funds or other sources of income may help. A college degree starts you as an E3 or E4...
So, talk to your recruiter and they will be able to give you more information...
I'm a single father with joint custody of 3 daughters and all I was required to do was get a notarized statement indicating that my daughters mother had full PHYSICAL custody.
Once I did that then the Battalion Commander called me and had a phone interview basically wanting to know about my finances. Just making sure I could survive on an E-4 salary because I would be taking a pay cut from my current job.
It was all approved in less then 24hrs.