what could it be?
it could anything down to a fuel filter. i know a gut who put so much money into his Z28 to find out a fuel problem. sounded just like your problem all he did was get good gas and change the fuel filter and it fixed it right up. im not saying thats whats wrong with yours but i would replace the fuel filter $5 and 5 min to install. i would try that its probably the cheapest fix.
its deffinatly not a lose connection the pump makes a faint electric pulse at the turn of the key. i can only hear this when im under the fuel tank and someone else turns the key. is this normal for a dead pump? or could it be a bad relay?
check the voltage at the rear where the pump plugs in. check your grounds too. if thats all good your pump is probably shot.
im pretty sure thats it but could the fuel pump be getting a messed up signal or act funny due to a bad relay? As i previously stated the fuel pump makes a buzzing noises that is only audible from under the car when someone else turns the key.
i would verify the voltage at the rear, check your grounds, then get a new pump if those check out. you should get clean battery voltage at the pump connector in the rear. likely, though, your pump is shot.
I doubt it would be a fault with the relay, it would be more then likely dirty/poor connections adding extra resistance to the circuit.
first, you will need some sort of multimeter, i prefer a digital volt/ohm one myself.
if the pump is turning on at all, it has continuity. it may have high resistance. you would want to check the resistance of the relay using an ohm meter. the diagram below shows the fuel pump relay. you would check the resistance between c1 and a2.

in addition, courtesy of shoebox, the below image has a pic of the harness at the back of the car. the wire labeled "fuel pump power" should have 12v+ with when the key is turned to the on position as it primes the fuel system. it should also be 12v+ if you put 12v+ to the fuel pump prime connector up front.
if the pump is turning on at all, it has continuity. it may have high resistance. you would want to check the resistance of the relay using an ohm meter. the diagram below shows the fuel pump relay. you would check the resistance between c1 and a2.

in addition, courtesy of shoebox, the below image has a pic of the harness at the back of the car. the wire labeled "fuel pump power" should have 12v+ with when the key is turned to the on position as it primes the fuel system. it should also be 12v+ if you put 12v+ to the fuel pump prime connector up front.


