First steps first:
Check your voltage.. multimeter (voltmeter) across the battery terminals. Started, and at idle should read 14-14.7 volts.
Rev the engine through about 3500-4000 rpm (via the TB from within the engine bay) and watch the readings... should maintain that voltage.
Shut the car off, and remove the surface charge of from the battery. (headlights and interiors on for 60 seconds)
Measure voltage at battery.. should be 12.0 or higher.
a measurement that is off while running would indicate alternator problems. A measurement of <12 while off would be a battery problem.
My guess would be that your wiring on the alternator is either corroded, or loose.
Ray
