You'll be working in a "damp" location, so you'll want to prevent moisture from entering or accumulating within the box. Since there's not likely to be moisture above the box, you probably won't have to worry about protecting the back side of the box. Because of this, you can probably get away with any type of box. When I did this, I used a weatherproof box and conduit.
National Electrical Code (NEC) calls for a cover that is weatherproof when a plug is not inserted for damp locations, however, I always use an "in-use" style cover on outside installations. I used this weatherproof expandable in-use cover which accordions out up to 3 1/2" when in use, and keeps a relatively low profile when not in use.
Obviously, you'll need ground-fault circuit protection. This can be provided either by a GFCI breaker, or GFCI receptacle.