Ok first we gotta fix that box. It has a missing knockout and does not have enough space. Fortunately you're in an unfinished part of the home so there's no drywall mudding to do.
- a nice, large 4" square steel box. It has a ground screw hole pre-tapped 10-32. Try to get one with a little bump for the ground screw so it won't be drilling into the wood.
- a lid for a steel box intended for one GFCI (meaning it has the rectangular Decora style opening).
- A #10-32 ground screw with or without pigtail.
- a foot of Romex of approprite size for pigtails (#12 if 20A breaker, otherwise #12 or #14)
- some wire nuts, always good to have.
Tear the old wires off the old box (it really won't matter) and pry the old box off the wall. On the new box, fit the ground screw in the hole tapped 10-32. This is mainly so when you're attaching it, you don't accidentally put a nail/screw in the ground hole.
I strongly recommend screws. Mount the new box. Now the ground screw must be able to bottom out, so do what you have to so it can. Put a ground pigtail on that ground screw. That is the key to the grounding system, you will not use the ground screw on the GFCI (so one less wire to wrangle, yay!)
Now knock out three of the knockouts on your 4" box for your existing two and one new cable. Use proper cable clamps like those on your old box. I would recommend doing one cable at a time.
On the GFCI/receptacle, leave the tape on the LOAD terminals.
Since you do not know which of your existing cables is supply, guess... and connect its hot and neutral to the LINE inputs of the GFCI/receptacle. Cap/wrap/position the other wires safely. Power up and see if the GFCI works. If not, try the other cable. So you have found supply. Leave it attached.
Now, fit your other cables into the box. Wire-nut all grounds together and push them into the back of the box, remember there is not a ground wire on the GFCI/receptacle.
Take the "know what you're doing" sticker off the LOAD terminals. Now there's only one hot screw (brass) and one neutral screw (silver) on the LOAD side, but you need to attach 2 wires to it. You can't stack wires, so you use a pigtail. That is an additional piece of wire you get from shucking the jacket off Romex, 6" should suffice. For each black and white, attach it to the appropriate LOAD terminal. Now you have 3 black wires, the old load, the new load, and the pigtail. Join them with a wire nut. Ditto white.
Or if you're not ready to light up the outside run, just leave it out (still wire-nut the pigtail to the old load wire) and add the new wire later. Protect the loose wires so they don't short against something, e.g. put a wire nut on the end and tape it down so it stays put (they like to come off single wires).
Button it all up: fit the receptacle to the lid, then the lid to the box.
Give it a test.
The receptacle gets its ground through surface contact yoke-lid and lid-box, then to the ground screw. External GFCI testers rely on ground as a fault path, so if you haven't buttoned it up and the tester hasn't worked, that is why.
A pigtail looks like this
Wait, why did they pigtail that, when they had enough screw terminals? Perhaps the location was awkward and they didn't want to be up there, back bent over, fitting 4 screws. So they fit the pigtails on the bench, in comfort, and crawled in there, 2 wirenuts, done! I do that all the time. Nothing wrong with using pigtails when you don't have to.