Sounds like either your alt is too small or not working properly. Your battery is not fully charged for one reason or another. Here is a few things to test, but first, take out the cap to rule it out.
1) Voltage at battery with engine off, key off
2) Voltage at battery with engine off, key on acc
3) Voltage at battery with engine on, all loads off
4) Voltage at battery with engine on, all loads on (headlights, audio, interior lights, heat blower, etc)
Take those readings and post back. I will be able to help you more then.