Until a real electrician comments, here's an opinion from an experienced homeowner...
Yeah, some kind of protection will be good to have for that wire. Either run it through the wall or through EMT conduit with water-tight fittings. How to go through the wall will depend upon where your enclosure is in relation to the existing GFCI outlet. Is there a stud in the way? Does the wiring for the existing outlet come through the wall or through EMT? How deep is the box that holds the existing outlet? (You'll need enough space to accommodate the new wire in that box. Always make connections inside boxes, not in the wall.) I'm not sure, but I think it's not a good idea to wire a second GFCI outlet after another one. So you might want to consider wiring the new one before, or in parallel with the existing.
Also, depending upon how much rain exposure you expect the new box to have, consider caulking around the top and sides (not the bottom) of the new box. This way rain will be kept from getting into your box or your wall.
So your idea could be pretty simple, or a fair amount of work, depending on your circumstances.
PS: Have you already considered just replacing the existing outlet with the new box?