Electrical – How to run romex around window with no studs

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.

I agree with @DA01 that it'll basically be impossible to fish horizontally through studs and insulation, without having to cut open all of the sheetrock. Then you have to deal with sealing it all back up air-tight, and patching the drywall and repainting (at least the entire wall, and maybe the entire room, depending the paint match).

So a couple alternatives:

Use a wiremold type product to surface mount the conduit:

There are lots of different brands, and it can be paintable. With planning, you can minimize where you see it.

Build up the wall using cabinets or a false wall, so you can run wire in behind, on top of the existing wall.

Could be as simple as just a drywall section sticking out, or some built-in cabinets or some brick work. Cabinets of course have the benefit of being able to build in other components such as your A/V gear and speakers.