Start with the basics. Pull the battery terminal ends, and clean them and the terminals till they shine. All it takes is a poor connection to give you what you got.
And the newer alternators are not designed like the old ones. They will not keep the engine running if there is no battery voltage. Don't try the old trick of jumping one and then pulling the battery cables off to check. New alternators have to see battery voltage to operate, if not they can go full output and fry all kinds of things, like your PCM.
Cables good and clean? Try to charge the battery, and then tool on down to AutoZone/Advance/PepBoys and have em run a check. It's free, and will check the battery, starter, and alternator to let you know if you have additional problems.
97 Sahara, 74 Renegade, 72 CJ5, 55 Willy's (the Beast) Boss 302, 6:34 gearing, Bose 901 speakers.
God I miss that Jeep. Forgot the kinda Jeep, 80 SX4.