5 years is nearing the end of any battery's normal life, no matter the warranty. Also, you would do well to ensure all power and ground connections are free of corrosion and tight. And don't just do a visual, because corrosion can be hidden out of sight between the connection.

My money's on the battery, though. And I'd be willing to bet the alternator wasn't bad to start with. If I were you, I'd see if the Carquest still has your core. Then, I'd pay them for it and shelve it for future use or to have rebuilt at an alternator shop...

Fasteddy's advice is occasionally sound...