By 'refurbish' are you referring to restoring the frame or base, replacing the interior padding, or recovering the cushions?
Is the recliner one that was originally intended for interior only use?
How much are you willing to spend on this project? Re-upholstering any furniture can be pricey.
If we see what style your recliner is, and what condition it is in now, we will be more able to offer suggestions on how it can be improved.
Please post a picture so that we can see what you are starting with.
Most indoor furniture, especially upholstered furniture, is not going to survive outside. More information would be helpful.
If a reg recliner for inside, it will be much cheaper to buy an exterior chair that reclines. Re redo for outside you have to gut the chair, reglue all joints with waterproof glue. Replace all stuffing with exterior foam and reupholster with exterior fabric.
You could make a cover for the chair with an outdoor tablecloth or shower curtain, but I'd be worried about bugs (and other critters) getting into the recliner to warm up when it gets cold. I would also worry about mold growing inside the chair if moisture gets inside.