I first started with a little hole to view inside for anything that would make installing a cabinet impossible. Such as a water pipe. It looked good from here so I continued breaking the drywall out.
I went slow and carefully because I did not want to hit any electrical wires that might be in the wall. As you can see I did find a stud, which I knew was there.
I then used a Jig Saw to remove the stud. This was tedious because I did not want to go through the wall so I had to cut slowly and chip away the stud.
This is a picture of the cabinet in the wall. I screwed into the stud on the left and into the top and bottom of the stud I cut. If you look closely the Medicine Cabinet particle board actually started to buckle because it is such a tight and secure attachment to the stud. (I'm not worried about the buckle at all.)