I'd just get the alternator and battery tested and go from there. I'd suspect something causing the alternator to put too much load on the engine, causing it to die. Could be it's trying to keep up with a bad battery, could be it's bad itself. But you'll never know unless you start testing things. Once it's started it should stay started, though.

A quick test would be to take the belt(s) off, start it, and see if it stays running then. But that wouldn't tell you much beyond what I said above and you'll still need to do the tests.

-Rog