Best way to run electrical wire behind built-in shelves, with the least patching afterwards?
You might consider using a surface-mount raceway.



You could use something like this to run the wires along the inside corners of your shelving up to the bumped out wall above the fireplace, then punch through into that cavity.

If you want to run HDMI (and maybe something like Ethernet), keep in mind that you cannot run line power and low-voltage cables together in the same conduit.

If you aren't going to run between three and five individual HDMI cables, also consider picking up an HDMI switch (with remote) so that you can plug all your devices into the switch, presumably sitting in the shelving, and run just one HDMI cable to feed the TV.



Having said all that, I tend to agree with the comment cautioning against putting the TV on the wall over a fireplace. The heat won't be friendly to the TV, and you might find that you don't like straining your neck to watch it.

In fact, you might consider mounting the TV up there temporarily and using extension cords for a week or two just to be sure you don't hate it. It would be a bummer to do all the work to permanently mount it there just to discover that you don't like it there.

Good luck!