Car electrical issues. Alternator? Battery?

Sounds like your alternator isn't charging the battery. When the car is "normal", you should see the idiot light for the alternator light up when you are starting the car.... when the key is turned to the start position and the engine is being turned by the starter. This tells you that the light bulb that is the alternator indicator is not burned out. That way, when you do have a problem with the alternator, the light will light and then you'll know that you have an alternator problem.