What is making my engine start then die

it could be a wiring problem in the circuit or the injectors have a resistance problem or maybe even the injector drivers in the computer have a problem or are bad.

It doesnt sound like you have sufficient fuel pressure can you hear the pump run? if so might be a pressure issue, if not probably pump

Most likely causes are bad engine coolant temp sensor OR leaking fuel injectors. Bad engine coolant temp sensor will give computer false reading and computer provides wrong air fuel mixture. Leaking fuel injectors will load the engine with raw gas--just like flooding it. Try pushing the gas pedal to the floor and holding it there when cranking when hot. That forces the computer into "clear flood" mode and it shuts off the injectors. If it starts up with pedal down, that would be a sign of leaking injectors.

It's a fused circuit, so check the fuse, probably under hood, labeled injector. A shorted injector can shut down the injector drive circuit, so pull off all injector connectors, make sure fuse is good, then check for power to the injectors. Injector circuit should nave hot wire anytime key is on. Has constant voltage to injectors. When the computer internally grounds the injector ground wire, the injector will pulse.

Check voltage at the injectors with key on. Voltage should always be available to injectors with key on. When the ecu grounds the injector circuit internally, for a split second, the injector will pulse, spraying fuel. You can buy a "noid" light to test injector circuit, one that fits your injector connector. To test, you crank the engine over, noid light should be blinking.
Or, use a test light and check injector drive circuit. With key on, one injector terminal should have constant battery voltage-test light should be lit bright. Now move ground clip of test light to the POSITIVE battery post, and probe the other injector terminal, the one that did not light the test light. Have someone crank the engine-if injector circuit is working, the test light will be blinking, indicating the ecu is grounding the injector circuit.
If no voltage to injectors, the drive circuit and wiring needs checked. If no ground signal, the wiring and the ecu needs checked.

Check if the injectors are pulsing when cranking the engine over. Listen for the tick, or feel for the pulse with your fingertips, or buy a cheap "noid light"-an injector circuit tester- one that fits your injector harness connectors-plug it into an injector connector and crank the car. If circuit is working the noid light will be flashing.
Some anti-theft systems immobilize the injector circuit when they are activated. So you have spark, compression, and fuel to the engine, check the injector circuit.

Have you checked the Cam sensor? The Cam sensor controls the injector pulse.

The ECM on this car will shut down the injector driver circuit if one of the six injectors has a coil out of tolerance. (Shorted coil or resistance out of parameters) Quick check. Unplug two wire cap to injector one at a time. (hold/push wire clip inwards then gently wiggle/lift cap off---careful with wires) With one unplugged try starting. If no start- plug back and use same procedure on next injector and so on.........In most cases if bad injector coil the engine will start when it is located. Replace injector. Some say the ECM shuts dirver circuit down to prevent further damage to switching circuit.

check on the wires to the automatic shut down relay are not shorting and also check out the Start relay circuit.

That sounds like a bad grounding wire, or possible PCM malfunction. Check all engine wiring harness locations for any corrosion or damage. let me know what engine type you have, and I could look up the wiring schematics