Battery or Alternator problem?

I have a 94 Chevy Silverado with about 16,000 on a rebuilt engine. Yesterday I went to out to go to the store and It wouldn't start. I jumped started it just fine, shut the engine off after a few minutes of idleing and it started up just fine. Went to the store, shut it off and when I came out it was dead. Got it jumped and drove home just fine and I let it run for about 20 minutes or so, shut it off and it won't start again. Today I pulled the cables off and tryed to clean them out to get a better connection. It started with out a jump but It sounded weak, drove it around the block and started it up and shut it down a few time with its sounding like a weak start. I came to your website and read on how to check the battery with a voltage tester. It was a 12.3. I then went to start it do see whats amps I got with it running to see if the alternator was ok and it would not start. So I guess my question is, with scraping the connection cables and battery to get a better connection and it starting weak from that and then dying again, Would this be a bad battery or alternator.

FYI... The battery is 5 years old and I tested the battery again after it wouldn't start and it was a 10.4 amps.

and

the lights, dome light and buzzer still work but it is slightly dime when the battery won't start the truck.
Thanks,