"Breach of Contract" on it's own might not be a crime, however some contracts may be enforced or protected by criminal laws here is a simple example your divorce settlement especially alimony (which may or may not include child support).

When you divorce some one you sign the divorce papers which are a contract certain parts of that contract can be enforced by filing criminal charges, as in the US at least if you don't pay what you affirmed to pay you can be charged with non-support which in many states is a felony.

http://www.ncsl.org/research/human-services/criminal-nonsupp...

You're confusing a breach of contract and the settlement of a suit. You sue for divorce and eventually reach a settlement. A settlement is enforced by the courts. (Having decided a breach exists and decided damages, the court can force a party to abide by the terms of the settlement.)