Wherever you want to put the outlet drill a small hole.
Run cable line from basement to first floor.
Run the wire or cable to its destination.
On the second floor run wires down the wall from the attic.
Attach the wire or cable to a fish tape or coat hanger using electrical tape and feed it into the hole until it reaches the opening at the other end.
If you have access to your basement crawl space and attic you should be able to run wires without having to cut patch any drywall.
Yes the cat6 from the basement to the upstairs is currently used to connect the ont to the router upstairs.
One will run next to a heat duct that goes up to the second floor.
Once you are all set simply shut the power off from the main board and get started.
When necessary use existing ducts or pipe runs to get wires from the attic into the basement.
On the first floor run wires up the wall from the basement.
When adding or replacing wiring in finished walls most electricians will attempt to make the horizontal portion of the cable run in the unfinished attic or basement crawlspace areas looping the cable down or up through the wall cavity across the floor or ceiling joists then vertically through another stud cavity to the next wall box opening.
If you re working from the attic you can drop a weighted string down wall cavity then attach the wire to the string and pull it through.
For the horizontal portion of the cable run this can involve drilling holes in the joists where the cable will pass.
From inside the living space position the bit against the wall directly above if accessing from attic or below if accessing from basement the desired location of the service the cable will provide.
Running through interior closet walls for all three.
Use the coat hanger to see if everything is fine at this location and there are no obstructions.