How do I tell if my alternator is charging?

Put a voltmeter right across the battery terminals while running. Should read vicinity of 13.5 to 14.5V at fast idle. With headlights on, should still stay above 13.0V

If those readings check, it's possible something is drawing current with the engine off. Shut it down and take the Negative cable off the battery. Connect a milliammeter from the cable end to the battery Neg post. If you read over ~50 milliamp, pull fuses/disconnect things until it goes down. Alarm systems or some stereos may draw a small residual current, but not enough to drain the battery.

Speedy Jim