Replaced outlet still not working
First thing to check, as @Tester101 points out, is that you've connected the  wires to the LINE terminals on the GFCI receptacle. It will also have LOAD terminals, which would go any other outlets you'd want fed from (and protected by) this GFCI. But it doesn't appear that you have any of those.

Next you should carefully use the contact terminals of your multi-meter to measure voltage on the lines:


If you measure around 120V between the black and white wires leading to the GFCI, and those wires go into the LINE terminals of the receptacle, the receptacle should be test/reset-able and should show 120V when you test between the plug-in slots on the front. If you have 120V at the LINE terminals and not at the front, the receptacle is defective.
If you don't have 120V between the black and white wires, your circuit is switched off or damaged. You'll need to investigate the wiring to determine what's wrong. Your non-contact tester may have given you a misleading indication; the contact tester is more authoritative here.
To identify the "true" neutral, measure voltage between an insulated wire and the bare ground wire. Neutral and safety-ground should have zero voltage, because they are bonded in your main electrical panel. So you should have 120V between black and white; 120V between black and ground; and 0V between white and ground. If you have 120V between white &amp; black but 0V on black/ground and 0V on white/ground, then your ground wire is not properly connected.


Finally, you should run a small length of bare wire between the ground bundle in this box and the ground screw on your switch, so that the switch's metal components are grounded.