My kinda DiY, sorta messy and make it up as you go: Is this a steel box fireplace surrounded by standard drywall construction in a newer home, or an older solid masonry one, which might have had a metal liner installed in the brick chimmney, and where the original walls would have been plaster over thin strips of wood lath? It's more common to take a brick chimney down to the roof than the double-wall sectional metal ones that hook to zero-clearance steel fireplaces.
Doug's note is very apropos. It's very important to know the rules made to keep you and the next folks in the house safe, especially if we're about to ignore them. With no chimney outside, you don't have a working fireplace, and should be safe.
I'd go for the box option, it's the right way to do it, and little more work or expense. The first wire that slips back into the wall, you'll wish you had. You want a plastic or metal double wide with a closed side (you're protecting from a short-lived fir in the box) for the 120VAC electrical and an open side for the low-voltage communications wires. You can bring those out with no cover plate, thru a simple hole in the cover plate, or wire then to a fancy plate with the real-deal connectors. Avoid a busy Saturday and the Home Despot guys will be happy to advise. The electrical side's a standard plug, which you buy: Black wire to brass screw, white to silver, ground to green.
1st Job: How do the wires get to the box? Your plan is to go in the fireplace face opening, up the throat, past the damper and out a hole to the box. If your fireplace is zero clearance, you still have an option to go fully concealed in the wall. First do a dry fit of the TV, it may come with a template or you could use the mount or cut a side from the box. Figure where you'll be happy and where your cables and connections have to be exit to be hidden. Pick a side off of dead centre—the flue is likely centre and you want some working room—for your cables to come out, based on which side's the easiest run. Use you finder so you're beside, not on a stud. There are boxes and clips that fasten boxes straight to the drywall, but screws into the stud are stronger.
2d: The Outlet: The carbide masonry bit in your hammer drill talked of above is the tool to drill the brick slices, and you'll need one installing the TV mount, but I'd just use it to chew away the mortar between the slices, then pry off enough to fit my box. Mrs castle will want a dropsheet, even if you don't. Save the pieces to patch with, and cut just a bit bigger (as in posts above) with a utility knife and/or drywall saw (a bread knife made fror carpentry) Be safe, take it slow. People who bully their knives wind up in ER. You should now be able to see the flue, and butcher your way through. See much info above. If you are facing bricks, you can use your masonry bits on the mortar and work one out, Then deal with the brick or flue that's likely behind it. Be sure to protect the cables from the sharp hole edges that are left, and don't make life harder with too small a hole. IF you see you're facing a void around a steel box, here's your chance to just punch a discrete hole in the dry wall, just above the baseboard and bring the cables past the firebox instead of through it. We invented plaster because it repairs so easily. Or buy a computer desktop cable flange at HD (Tools and hardware dept), and just leave the hole.
3d Wiring: Mount the box, as described. Use a string from your hole to fish the bundle of cables. Tie a small heavy load to it, push iy=t into the flue and let it drop until the damper stops it, at which point you'll have to go pullit past without your end getting pulled into the hole—it's why they call it the bitter end of the anchor chain—tie securely onto the cable bundle, maybe torpedo the blunt end into something that'll run easier with some tape and pull them through.
Secure each to its box with at least a healthy 6" of 'pigtail to work with if your making connections like electrical there. If the cables will go all the way to the TV, the box is to make sure they do not go back into the wall, give yourself the required length, secure and wire any connections. Go back along the wiring path to tidy up and secure every thing. Patch around the box with the brick pieces (which can be snapped, although its hard to be accurate. A score line will help)
4 Install the TV mount as per instructions—which should say what size carbide masonry bit you need.
Enjoy. And if I ever discover you're actually cute and petite with perky B's and I didn't volunteer, you're toast frank.
