I recently got an 86 IROC with a crate 350 in it. It sat around awhile (not sure how long exactly), and it's been driven very infrequently until it came into my hands. I drove the vehicle for about a week and all was okay, but now it seems as if the battery is not getting charged. The lights start to go dim while I'm driving, using the brakes, shifting, etc and so I attributed these to the alternator, which I then replaced. The car still does this even after I replaced the alternator and checked the wires from the alternator to the battery. To describe what happens more, the battery seems to get weaker the longer I drive, and if I shut the car off and then go to restart it after about 5-10 minutes of driving, it sounds as if the battery is completely worn out. If I come back half and hour later and try it, it will start right up, but the light issues persist. Any ideas?

PS. I had the battery tested at autozone and they think it's okay, although I think it is undersized for the engine.

 

Try getting a larger battery, more cca won't hurt.
Although I have a 350 swap in my 86 Iroc and I am using the original size battery (its been replaced)

And what do you mean by "it sounds like the battery is completely worn out"?
Check your grounds and battery terminals, Clean them all off.

 

Check the voltage at the battery while the car is running with a multimeter.

If the alternator is not charging, check the voltage and both pins on the plug on the side of the alternator. With the key on, both should have 12 volts.