Is there a way to pay my credit card balance with bitcoins?
If you live in the states the only service that comes close to this would be

https://www.xmlgold.eu/  - you can even order a pre-paid credit card that you are able to top up with bitcoins.

https://www.okpay.com/en/index.html - Okpay used to offer the same service but due to regulatory reasons they have ceased offering thier services to US citizens.  

The only other way I can think of to do this would be to use http://bitcoin-otc.com/  here you  can see if you can find anyone willing to pay off your credit card for bitcoins.  Its kind of like online trading pit.  Hope this helps.