I'm a US citizen marrying a German citizen and want to move to Germany to live but I have a criminal record an owe child support in the US. Is there a way to accomplish this?
A criminal record is actually less of an issue for spousal visas than for any other type of visa so it's definitely possible to accomplish this. The reason for that is that the courts have to balance the interests of society in general against the right to a family life, which is also strongly protected.
Therefore, the mere fact you have a record does not automatically disqualify you, it will come down to the severity of the crime and any other indication that you currently present a danger to society. I don't actually know this for a fact but I also suspect that paying what you owe would help make your case stronger (unlike a record, not paying child support speaks to your current conduct) so settle any debt you may have and make sure not to miss any payment, as applicable.
Beyond that, I would urge you to instruct a lawyer with a practice in German immigration law. You have to be very careful about the way you argue your case and random advice you get on the web can only bring you so far in complex situations like this one.