Replace existing phone jack with network cable
Go fishing
If you're a really lucky person, you might be able to use a fish tape to follow the path of the phone cable.  However, this means running the fish tape through the wall or ceiling, and hitting a small hole in the framing.  Without being able to see what you're doing, you'll just be poking around in the dark, trying to hit a hole of unknown size.

If you want to attempt this, the procedure is as follows:

Run the fish tape up the wall, or down the ceiling cavity.
Poke around until it slides into the hole through the top plate.
continue to feed the fish tape, until it pops out the other end.
Attach the new cable to the end of the fish tape.
Pull the fish tape back out the way it went in.

Open access
The &quot;easier&quot; option, is to cut one or two small access holes. This will allow you to drill new hole(s) through the framing members, and let you see what you're doing while running the cable.
While running the cable, you're going to encounter at least one framing member. Right at the top of the wall, there will be a top plate. Which will consist of either a single, or doubled up piece of lumber laying flat across the top of the wall. In some situations, you might also have fireblocking, or other obstructions within the wall. You'll want to use a stud finder, to try and locate an obstructions along the route.
If you're only dealing with a top plate, you'll want to cut an access hole just below/above the corner of the room.

This hole should be just large enough to fit a flexible drill bit, to allow you to drill through the top plate.

Once you have the hole through the top plate, fishing the cable is easy.

Feed a fish tape up from the lower hole in the wall, to the upper hole.
Feed the fish tape up through the hole in the top plate.
Feed the fish tape through the ceiling, until it comes out where you have access.
Attach the cable to the fish tape.
Retrieve the fish tape and cable.
Patch the hole(s) in the wall/ceiling.