Hows your carpentry skills Ian? We're about to learn a little about framing...
First there is no problem with drilling down if it is a bearing wall - that is if they match up, and pretty sure from here
that they will. But it would be wise to measure from outside walls on each floor to make sure - even off by an inch or two you should be OK.
You can also safely assume that this is platform framing, as it is not a Victorian.... Or built in the 1890's...
Either way one can bet that the joists 8 of 10X's that the joist will run in the short dimention across the house. e.g. if the house is a rectangle - it would be that short dimention. And the joists will run accross the wall you plan to go down by looking at your sketch. Often - but not always the studs and joist will line up on the first floor, and often fairly close to lining up on the second floor. That said if you open up a stud bay in the closet up stairs you should be right above the wall you want to get into.
Measure from the front of the house on each floor to see if the area you want the tv at is below or close to the closet above, or if it ends up in the bedroom...
If in the closet... Use a good and powerfull magnet to find the screws that hold the sheet-rock on the wall, and that will tell you were the studs are... Take a level, or a plumb-bob and put a line up-down on the wall where the studs are on each side of the stud bay. And open the steetrock from the center of each stud in the bay you want from about 4" above the baseboard to ~20" high. Carefully remove the piece of rock, so you can put it back.... (You'll crack an edge - don't worry...) This way you can fit a drill into the wall.
Put a mark or tape on a 7/8" spade bit about 4 1/2" from the flat cutters, and drill through the bottom (two) plates and the sub-floor in the middle of the stud-bay. Carefull to go slower at the mark. If you go past the mark you are on the joist - do not continue... Move over a few inches and try again... This will put you in the joist-bay, and you should be able to peer down with a flashlight to see the top-plate of the wall below. Drill trough the center of that with a 1/2" x 18" bit. Then you are in the stud bay of the wall below. Your kitchen is below that.... So be carefull, there may be other things in the wall...
Try to drop a jack-chain, or a bunch of 1/4-20 nuts on a string down the wall, if there is plumbing or worse blocking... Then its plan B...
If not run surface in the closet, and into you hole in the closet down to the TV loc...
Anyway, you should do some more planning and take it slow if you do this yourself.