Sorry, I don't understand your project at all!
You can control an outlet with a relay. There are just a couple of issues to be aware of -
The Arduino can't directly-drive a relay coil, so you either have to make a simple transistor/MOSFET driver circuit, or you can buy relay boards with the driver built-in.
The relay coil generally needs it's own separate power supply (or you should have separate voltage regulators for the Arduino and relay). Or, if you use a 12V coil and a 12V power supply the Arduino's on-board 5V regulator may provide enough "isolation" to prevent glitches when the relay switches.
There isn't usually enough room inside the electrical box to put the relay inside. If you want to control a single-outlet, a double-outlet box gives you extra space (and you can get an outlet-cover with on side blank). I have made some small "homemade" solid state relays with an opto-isolator and TRIAC, and jammed 4 of them into a quad outlet box, but it's a mess!
And it might be illegal (a building code violation) to mix low-voltage and line voltage in an electrical box that's part of a structure.
You can get solid state relays that CAN be directly driven by the 5V low-current output from an Arduino. These are super-easy to use and if you get the [u]industrial type[/u] with screw terminals, they are super-easy to wire up.
Depending on your project you might look-into some [u]home automation[/u] products. You can buy wirelessly-controlled outlets (and switches/dimmers) that fit-into a standard outlet box.
I don't get what GFCI has to do with this, unless you want to "hack" it to use the internal relay? That doesn't seem like a "smart" thing to do...
GFCI has a particular purpose... It senses current through ground and cut-off power. So if you are drying your hair while taking a bath and drop the hair dryer in the water, it shuts-off power when it senses current though water to ground, hopefully in time for your heart to re-start.
I've been told that you can wire multiple outlets to the GFCI so all of the following outlets down the string are also protected. That makes sense... One GFCI outlet can protect the whole kitchen or the whole bathroom.