It definitely sounds like it could be your alternator. However, the first thing I'd check is the alternator belt. Make sure that it's tensioned correctly and isn't worn. Then check that the wiring harness for the alternator is secured tightly, and not burnt or cut in any way. If everything checks out ok then it's time to test the alternator with a multimeter.
With the engine off, connect the multimeter to the battery terminals and make sure you have at least 12 volts. Now start the engine and check the battery voltage again. It should be between 13.5 to 15.1 volts. Now turn on the headlights. The voltage should drop and then come back up if the charging system is working properly.
I'd recommend taking the car to an auto electrician and getting the problem professionally diagnosed. Most places do the diagnosis for free. They have all the right equipment to do proper tests (such as an amperage output test) on the charging system. Good luck!