It's possible that it's starting on the Cold Start Injector and then dying because the fuel pump shuts off after you release the key to the "run" position. Once the engine starts, the fuel pump is supposed to be energized through the Circuit Opening Relay, which is turned on by a switch in the Air Flow Meter. Look at the circuit below to see how it's wired.
You can test to see if the fuel pump is running all right by making a jumper out of a paper clip and jumping the Fuel Pump Test Connector, on the drivers fenderwell like this:
With the jumper installed and the ignition key "on" you should hear the fuel pump running and you should hear fuel flowing through the fuel rail. Try to start it now with the jumper installed. If it runs now, there's a problem in the switch in the AFM or the wiring to the COR.