Open the Stud Cavities, and Push Out the Wall
Once the wallboards are off on one side of the wall, you can attack the other side of the wall. If it's an interior wall and it's wallboard on the other side, repeat the process you just completed.
If the other side of the wall is an exterior surface, first check outside to make sure you won't cut anything you're not supposed to. On this project the exterior wall is wood. When you're sure that it is safe to cut, make a plunge cut through the wall with the reciprocating saw (Image 1) and a standard bi-metal blade. Make a cut from top to bottom along the studs in each stud cavity.
After cutting through the boards, just push the wall out from one side. Work the boards free as well as any excess pieces nailed directly to the studs. Be careful not to cut any electrical wires or boxes (Image 2).
You will also need to remove drywall from the ceiling area where the new header supports the ceiling rafters. Use a utility knife to score the drywall and use either a keyhole saw or a reciprocating saw to cut away the drywall. You may need to get into the attic space to insure you won't be hitting any electrical wires in the attic.