Drywall – How to finish a hole in the wall

If you're just using a regular outlet as you suggest behind the TV, then my suggestion would be to supply it like a regular outlet.

If there is another outlet nearby (directly below would be ideal), fish a wire to it, and attach it to that circuit.

If there is unfinished space below, you can get a wire down there (use a long "installer's drill bit") and then it should be relatively easy to find a circuit to attach it to (or just run it back to the panel).

I'd be nervous about any contraptions that attempt to get power to this circuit otherwise. You absolutely need to have a male end in the wall, as a male-to-male extension cable is dangerous to have.

The best thing I can think up totally as a DIY without requiring real hard-wiring would be to get a blank faceplate, and put a square hole in it, and then mount an IEC 320-C14 connector in it (eg. from digikey).

Then you can just use a standard C13 cord (like the kind used to connect most computers, TVs, etc) and plug it into a surge protector or UPS or whatever.

Get yourself some shelf brackets and attach them to your wall into studs. Work out the spacing of the brackets for your screen. The brackets look like this:

Attach some 3/4" hardwood (oak, maple, etc.) planks to it to make a nice looking shelf. Attach the screen underneath the shelves which should conceal the hanging brackets and give you the clearance away from the wall you're looking for. You may want to drill a hole through the shelves and attach the screen using bolts and nuts instead of screwing directly into the wood shelves. You can also use the shelf to display some cool movie knickknacks.

You could also build something fancy using a French Cleat if you have the skills and tools.