just had battery/alternator problems on two cars a few months ago. This is how I'd proceed:
1) put a battery charger on the battery. 2 amp for 12 hours or 10 amp for 4 hours.
2) take a digital volt meter and check the voltage on the battery - it should be above 12.4 volts. If it is lower - remove the positive terminal from the battery and check again. If the voltage goes up - you have an electrical problem - if it does not - you need another new battery.
3) once you have 12.4 volts or higher on the battery with the positive terminal hooked up - start the car. Check the voltage on the battery again it should be 13.5 volts Minimum to 14.7 volts. - If not you have alternator problems
Try these and report back and I can tell you the next steps.