It sounds like you suspect the fuel pump relay. I guess this is as good a place to start as if you test here first you can rule out everything up to the engine compartment. Remove the fuel pump relay, I believe it is the number 2 relay for your car but use a manual or search the internet first to be sure, and place a jumper wire between the battery power terminal and the ground terminal. These will be the larger terminals from the five which should have contacts in them, again see a manual to be sure of what you are doing. The wire should be of a thicker size, 14ga or larger should be fine, and a fuse would be a good idea but most people don't fuse it when testing. Then switch the ignition on, do you hear the pumps running? No, then you have electrical problems to track down. Yes, then turn the key the rest of the way to try and start the engine. Does it start and run? No, then it's time to switch to a different troubleshooting procedure. Yes, then your current relay is/going bad and you should replace it (you should not drive around with the jumper wire installed).
You are not real clear if the engine is just not starting, dying right after it starts or dying a short time after it starts. So the above might not really apply but in the mean time it can't hurt to do the testing if you care to.