how to route cable in wall
It's going to depend on the type of framing used. Balloon and platform framing are the two common methods.

Platform
If you're dealing with platform framing, you're going to have to drill through quite a bit to go between floors.  First you'll hit the bottom plate of the upper wall, followed by the decking (subfloor). Next you'll hit a double top plate, which once you're through you'll be in the lower wall.  Since it's a common wall, you'll also likely have to deal with blocking and insulation.
Balloon
If it's a balloon framed structure, you're lucky. You'll likely only encounter blocking, bracing, and insulation. However, these elements are a bit less predictable, so you'll have to take things as they come.
Fireblocking &amp; Draftstopping
Any time you create a passageway for fire or gases to pass between floors, you'll have to provide adequate fireblocking and/or draftstopping. This could be as minimal as some fire resistant spray foam, but you'll have to check with your local government to determine what's required.