First, mad props to Sdiver2489 for recommending to put all wires on the LINE terminals. LOAD should never be used for any purpose except providing GFCI protection to downline parts of the circuit. And you should never do that unless you know what those downline parts are (obviously once you think about it) and also it makes sense to provide GFCI protection to them.
I want you to do exactly that. Obviously power doesn't end here, it goes onward to other points of use. I want you to suss them out. If you turn this breaker off (you *did* turn the breaker off, *right?*), what else loses power? Does it seem likely that it's farther down the chain from this recep location?
I also want to know what's upline. Are there any outlets indoors served by this circuit?
Because I've had bad luck with GFCI receps left outdoors. I much prefer to have the GFCI in the nice dry warm inside.
So I'm fishing for an opportunity to put the GFCI in an indoor location, and use its LOAD terminals to protect this outdoor downline. If you can find (or create) a spot where power comes from the panel, to an indoor recep location, to this outdoor recep location, to other places etc... then put the GFCI indoors and use LOAD.
A "Plan B" is to fit a GFCI breaker, which is all LOAD line obviously since it doesn't have any receps of its own. You won't like the price. For Pete's sake get a breaker that is correct for your panel (BR inside BR panels, HOM inside HOM panels, etc. Don't be "That guy" who sticks three $90 BR 2-pole breakers in a Siemens panel and has to replace em all.)
If you can do that, follow sdiver's advice but use a plain WR (Weather Resistant) outlet, and fit an outdoor in-use cover. (this is one of the RARE cases where I think that Code requirement is bullpucky, because I find outdoor in-use covers do not perform well and are a waste of money. We use them because Code requires, but we put them inside NEMA 3R enclosures, which keeps the rain off but the GFCIs still fail.)