Can I add a receptacle from a bath fan switch box?
Assuming your fan switch is wired normally, it has the following wires:

Hot (black) - this is coming in and should be connected to power in the fan box (but not connected to the fan)
White (neutral) - this is coming in and is not currently used - but you will use it shortly.
Red (switched hot) - this is going out to the fan.

Trust, but Verify
You should first verify that black and red are wired correctly. We can be fairly certain that white is indeed neutral because it is capped off, which is normal for a modern basic switch installation.

Turn off the breaker.
Open up the switch and disconnect (and pull apart for safety) the wires
Turn on the breaker and check that you have 120V (nominal) on the black wire. You can check with a non-contact tester (should light up next to the black wire but not next to the red or white wire) and with a multimeter (should show 120V between black and white and 0V between red and white). If black and red are swapped then you can still do this installation but it will be more confusing.

Install the new Cable
The new wire should be 12/2. You may only need 15A but since you (a) are installing a dual receptacle so you CAN install it on a 20A circuit and (b) if you changed to 14/2 you would have to replace the 20A breaker that is likely in place (since all the existing wiring is 12), stick with 12 so you don't have to change the breaker.
These instructions assume the black wire is hot.
In the switch box:

Turn off the breaker.
Wire nut together the original black wire, the new black wire, and a short piece of black wire (aka pigtail).
Wire nut together the original white wire and the new white wire.
Wire nut together the original ground (maybe green, probably bare) wire, the new ground and a pigtail (green or bare)
Connect the switch to the black pigtail, the ground pigtail and the red wire.

Run the new cable to the new box. In the new box:

Connect black to the Line hot screw
Connect white to the Line neutral screw
Connect ground to the ground screw

Do not use the Load screws on the GFCI receptacle at this time.
Then turn on the breaker and make sure the switch/fan and new receptacles all work.