How do I add an outlet between two other outlets?
I would run up to the attic, over, and down. Unless you have basement access, it's the only way you'll do this without cutting up the wall. (I also value the absence of cords across my walls.)


Carefully disconnect all wires in one box, removing the outlet and all nuts. Take photos or make notes if you think you'll need them later.
Depending on the box type, remove it by cutting the nails with a small saw. A hacksaw blade in a Vice Grip plier can be useful. If it's a brown fiber box you can break it up with a plier. Remove the box from the wall completely while taking care to not damage your drywall. A couple layers of painter's tape around the opening are a good idea. 
From the attic, drill through the double top plate in the same stud bay as the box. 
Drop a new 12/2 cable down to the box opening, ideally in front of the insulation but behind the vapor barrier. An assistant is helpful here.
Cut a new hole behind your tv, sized for a remodeler (old work) box. Double check stud locations first. You don't want to have to notch a stud. If you mess up there, a sharp 1" spade bit will make quick work of the stud. Be sure to consider your tv mount location as well.
Drill through the top plates in that stud bay as well, and drop other end of the cable down.
Install remodeler boxes at both locations and install the outlets. Be sure to use a deeper box at the original location as it'll have many connections.


Be prepared to do some sweating and swearing, but for the aesthetically-minded homeowner it's worth it to have completely clean walls around a hanging tv panel.