I'm guessing you have at least the height of the floor joists along the ext
wall in the basement to "work" and can see up to the bottom of the floor.
Do you just need to go a foot or two up from the floor into the kitchen wall
where a "normal" tv cable outlet would be?
Otherwise, if you're going up to above "counter height" for a TV/monitor
(say, mounted on a cabinet etc) you would run into fireblocking.
Also, if the home has a vapor barrier on the exterior walls, you should try
to avoid damaging it and the insulation.

If the desired location is standard (just about 16" above the kitchen
floor), what I've done is mark the location in the kitchen (after using a
stud finder) ..find the corresponding space between the basement studs and
ensure theres nothing there a drill bit will hit, then cut a hole in the
kitchenwall the size of a standard "EZ" box.

Then (I have two installer's drill bits..one is 3' long and the other is 6'
long. They are a long flexible shaft with a pointed "spade" type bit on the
end)
I'd take the 3 footer, make a small hole in the vapor barrier and carefully
feed the drillbit DOWN (by hand) into the wall to the bottom plate thru the
hole you' ve cut. Measure along the shaft to make sure you're "resting" on
the plate and not on something else that may be in the wall. The drill-end
of the bit would be sticking out of the hole, against the sheetrock at about
20-30 degrees due to its natural curve and the 'bit-end' should be almost
vertical and perpendicular to/against the plate. Apply drill to bit and
drill the hole while a 'helper' keeps an eye out for the bit in the
basement.

Going in this way (down) keeps the damage to the vapor barrier/insulation
to a minimum. Once the hole is drilled, connect a thin piece of cord to the
drillbit (it has a hole thru it just for that purpose) and pull the cord/
wire back up thru the hole and into the kitchen. You may be able to rent
such a bit or buy one at HD/Lowes etc.

I want to run/add a television cable from the already existing terminal
located in the basement, up through the floor to the kitchen inside the
exterior wall. The same as you would for any other type of outlet (phone,
switch, etc.) through the floor and inside an interior wall such as a
bedroom. Not running a cable from the outside to inside, but all inside.
I have run cables, phone lines, electrical wiring, etc. through many
other walls in this house as well as others, but never inside a wall
which is the exterior wall of a house. Therefore, I have no idea if
there's much of a difference.