GFCI outlet wired but not working
Put the tape back on the LOAD terminals.   You may know that common receptacles have two screws on each side, one is a spare for continuing the circuit(or for splitting the receptacle for controlling each socket separately; watch out for a broken tab between screws.)  The LOAD terminals are not spares.  They have a specific purpose and we won't be using them for awhile.  

Now, connecting only to the two LINE screws, hook up the hot and neutral from one cable.  Aim for it to be from the power supply.  Does the GFCI power up and act normally? 

If not, the cable you used is probably not the supply cable.  Try another.

Okay, you now have a working GFCI.  



Now the rest of the wires.  Consider these questions: 


Are you sure the supply cable has only 2 wires (and a ground) with no red wire? 
Do you know where the other wires go? 
Do you want to provide GFCI protection to those other places? 
Are you sure those places are properly wired and do not have any ground faults? 
Are you ok with this GFCI tripping if one of those places has a subtle electrical problem? 
If that happens, are you prepared to troubleshoot it? 


If any of those questions are "no", then leave the tape on the GFCI "Load" terminals as you will not be using them today.  

Use the "pigtail" techique - attach short wires to the GFCI's "Line" terminals, and use wire-nuts to join all the always-hot wires together and all the neutral wires together which are part of this circuit.  



If the answer to all the above questions is "yes", then leave the wires currently on the LINE terminals where they are.  Here we go!  Peel the tape off the LOAD terminals.  Attach the remaining always-hot wire(s) to the brass and the remaining neutral wires to the silver.   

Turn the breaker back on (it goes without saying that you do ALL wire changes with the breaker turned off) and run the GFCI through its paces.  Then go to all those downline destinations and test them too.  Does the GFCI trip?  NO? Then you are done.  

Yes?  Then there is a problem with the downline loads.  Fix it.   Or if you are not willing to fix it, then one of your answers above should have been "no" and put the tape back on the LOAD terminals and wire it as above.