How to check if my battery is OK?
I think what matters most is the amperage of your charger. If you are plugging your phone into a computer then it is limited to 500mA. A charger for the S3 will typically provide 1A to 2A I believe.

One more thing which may be going wrong is battery hogging applications. If you switch your phone off and then charge, by how much does your charging time lessen down?