How do I mount a TV to cover a cubby hole above my fireplace?
I would build a 2x4 (or 2x6, it's hard to tell the depth from this picture) frame to fit into the cubby hole, something like this:



Spacing and attaching

Space the studs 16" on center. You'll need to attach this securely to the rest of the walls. Assuming they're also wood, a few 3" #10 screws into either side would probably do the trick. Be sure to recess the frame by 1/2" so when you put drywall on it's flush.

Blocking

The blocking can just be a 2x4 or 2x6 flat against the front edge, and gives you a very solid surface to screw a mount into. You probably only need it for one edge of the mount, so long as you screw the bottom into at least one stud its should be sturdy enough. 

You can also use plywood (or OSB) on the surface, either with a layer of drywall overtop or just in the center of the hole, butting the plywood edge to the drywall. Kind of your choice which way is better, just remember if the finish is all drywall, it's easier to remove the TV and just patch the holes later. 

Power

It looks like there is already power there. Keep in mind all junctions MUST be accessible from the finished space, so this means you'll have to put power in a spot that the existing wiring can reach, or re-run wiring. There are various types of recessed plugs that can make things easier when dealing with a mounted TV.



Low-voltage signal cable

Some of the plugs also have low-voltage, or you can run that separately. Again, I don't know what's there now, so I can't make specific recommendations. It's not against code to make signal wire connections in the wall (eg, splicing coax), it's just not necessarily the best idea because if the connection fails, it's not accessible. 




I'm a fan of just having the cables come out of the wall (without separate connection jacks) behind the TV since 1) it's cheaper, 2) it's easier to install, 3) it's less likely to fail. The picture above on the left is one way to do this; in my last house I used a recessed media box that has grommets on the faceplate, and this is a great way to go:



(Note, I did that nearly 5 years ago, and ran everything I thought I might possibly need. I only ever used a single HDMI cable, and actually eventually ended up pulling most of the other cables out. Since I had everything in two 2" central vac tube conduits going straight about 10' opening into an unfinished space, this was trivial for me. YMMV.)

You may also want to run cables to a spot on the wall next to the fireplace, to put your sources (receiver, game console, whatever) there instead of on the mantle. Since you're going to be patching drywall anyway, if you need to punch some holes to fish wires, now is a good time to do it.

Conduit (optional)

You might also consider running a conduit, so you can run future cables through. If you do this, be sure to run a reasonably big conduit; cables like HDMI can't really be spliced so you have to able to fit the end through. Also, don't run anything in the conduit today (except for a pull string, if you want). Just run existing cables beside the conduit; if you end up having to replace them the cables are obsolete anyway, why bother with trying to pull them out and potentially getting stuck, etc. 

Drywall / Paint

Finally, put new drywall up and patch it in, feathering the edges (there are lots of drywall questions here to address that part, specifically). This will typically take 2-3 days since you have to wait between coats. 

Finally, you can paint. If you have the original paint and it was painted fairly recently, you MAY be able to just paint the new part and have it blend without being noticeable. More likely, you'll have to paint this entire surface (up to any edges/corners), so be prepared for that. Around corners even reasonably big differences in color are not perceptible, but on a flat surface even the slightest color difference is very obvious (trust me, I've been there, including the "Crap.. maybe it's just because it's still wet" thought, to which the answer is: no, it will still look off when it dries). 

If you don't have the original paint you can color match. Also, obviously, if you've been thinking of changing the room color anyway, now is a good time. 



Good luck, let us know how it turns out