if you turn the key on the fuel pump is only supposed to get power for about 2 seconds. if you crank engine over then it will run as long as it seeing a signal from the ignition system. need to have someone crank engine over as you check for power coming out of the relay or at the wires under rear seat going to the fuel pump. good thing about it is you do not need to remove the gas tank to replace fuel pump. here is a video on how to replace it and to get to wires going to it
https://www.youtube.com/watch?v=MWJREx7yn3w