I've been looking into this too, so here is what I have figured out so far.
1. DIY? I've installed a number of outlets, lights, switches, etc., have the tools, and feel comfortable doing it. If you've never touched electrical before you might consider having a friend who has done so help you. It's not hard but you have to do it right. Don't cut corners. My philosophy is to overdo everything, double and triple check, etc. belt and suspenders all the way.
2. I'd do a 20 amp GFCI. Outdoors GFCI is required. If you're running wire anyway you might as well do 20 amp. And if you ever decide to use a welder or compressor or other big equipment back there (or a future contractor/owner/etc. might) it will be nice to have. Get a 20 amp WR rated outlet.
3. 12 gauge solid THHN (individual conductors) in rigid PVC or EMT conduit. No Romex is allowed outdoors/underground/in wet areas, and 12-2 Romex in 1/2 EMT is pretty close to the maximum fill anyway. I guess you could use UF-rated cable in enclosed spaces but conduit is still needed for mechanical protection where it is exposed. Watertight conduit fittings outside, caulk around any conduit that runs through a wall.
4. Exterior box with an "extra duty in-use" cover is code. I have brick, but on vinyl here is what I would do: Cut a hole in the siding, caulk (top and sides, not the bottom) and screw the box to the sheathing, and put J-channel around the box.
Tug on all your connections to make sure they are secure. Put a couple wraps of electrical tape around your outlet and wire nuts to make sure everything is held properly in place.