It's doable. The outlet will need to GFCI protected. Any electrician can do it.

How you did it would depend on how the medicine cabinet was mounted, the cabinet depth, and the available space inside the wall around it. If the cabinet were recessed into the wall and there were space on either the side or bottom, you could probably install an electrical box inside the wall with an outlet that protruded through the cabinet. If the wall were thick or the cabinet were surface mounted, you might be able to put the electrical box and outlet behind the cabinet so the outlet protruded through the back of the cabinet. Worst case, you could just cut a hole in the cabinet for the wiring and put a surface-mounted outlet (example) inside the cabinet.