by relay you mean the soleniod mounted in the engine bay by the battery that the cables mounted to? you could have gotten a bad one, you can jump it. did you replace the starter itself or atleast pull it and have it bench tested, it may be junk.
Sounds like something isnt making a good connection..my experience like yours turned out to be the battery cables. Check the wires down at the starter also.
Since some of the tests will bypass the safety interlocks, make sure that the car is in neutral and the parking brake is set. Becoming a pancake isn’t part of the repair process…
Check battery, terminal connections, ground, starter relay switch (also known as solenoid) and starter in that order. The clamp on with 2 bolts battery terminal ends are a know problem causer.
A voltmeter is handy if you are familiar with how to use it to find bad connections. Measure the voltage drop across a connection while trying to start the car: more than .5 volts across a connection indicates a problem. See http://www.fluke.com/application_notes/automotive/circuit.asp?AGID=1&SID=103 for help for help troubleshooting voltage drops across grounds.
1.) Will the car start if it is jumped? Then clean battery terminals and check battery.
2.) Check the battery to engine block ground, and the ground behind the engine to the firewall.
3.) Jump the big terminals on the starter relay next to the battery with a screwdriver - watch out for the sparks! If the engine cranks, the starter and power wiring is good. The starter relay is also known as a starter solenoid.
4.) Then pull the small push on connector (small red/blue wire) off the starter solenoid (Looks like it is stuck on a screw). Then jump between the screw and the terminal that is connected to the battery. If it starts, the relay is good and your problem is in the rest of the circuit.
5.) Remember to check the ignition switch, neutral safety switch on auto trans and the clutch safety switch on manual trans cars. If they are good, then you have wiring problems.
Typical start circuit... Diagram courtesy of Tmoss &Stang&2birds
6.) The starter may be hung, loosen up the bolts that hold it on, and give it a good whack with a big hammer. Tighten up the bolts and try again.
7.) If that doesn't work, use a jumper cable from the positive lead on the battery direct to the starter post where the big wire from the relay connects. If it cranks then, it is the power wire from the relay gone bad. This will be hard to do, since there isn't much room to do it.
8.) Pull the starter and take it to Autozone or Pep Boys and have them test it. Starter fails test, then replace it. If you got this far, the starter is probably bad.