Outlet has ground wire connected to neutral and I am trying to replace it with GFCI
The good news is that a properly installed GFCI effectively protects you not only from the ground faults that it is designed for, but also makes things so that the key use of the ground wire - to provide an alternate path for return current should there be certain types of failures in a device - will effectively be unnecessary because those same faults will also trip the GFCI. So safety-wise you'll be fine.
FYI, if you have a metal box, it may actually be grounded after all. You can test that by seeing if you get continuity between neutral and the bare metal box. If you do, then you can connect a ground wire to the GFCI receptacle ground screw and screw it into the box. And if the GFCI is &quot;self grounding&quot; then you won't even need that ground wire at all. But if you don't have that ground, you are still safe with a GFCI.
The key with a GFCI is to know which hot (black) and neutral (white) are LINE and which ones are LOAD. You have, as is typical, two hot wires and two white wires. One of those sets is coming from the breaker panel (line) and the other is feeding the next receptacle (load). If you know which is which, great. If you don't know:

Turn off the breaker
Separate the hot wires
Turn on the breaker
Using a non-contact tester like this one:


determine which black wire is actually hot. That is the LINE hot wire. The matching white wire is the load neutral wire. If you can't figure out which black &amp; white wires are paired then it gets a little more complicated (but there are ways to figure it out).
Now that you have determined line vs. load:

Turn off the breaker
Connect the LINE wires only to the GFCI. The screws should be clearly labeled on the GFCI. (Hint: The LOAD screws are the ones that were originally covered with a piece of tape.)
Turn on the breaker
Test the GFCI - Test/Reset, plug in a device (lamp, radio, whatever) and Test again, etc.

If everything is good:

Turn off the breaker
Connect the LOAD wires to the other screws.
Turn on the breaker
Test the GFCI - everything should work now unless there is a wiring fault at one of the other receptacles.
Test the other receptacles.