Yeah you can, you have to ....."win the 100cc Special Cup. Alternately, win 1,650 races."
To get a second outfit for your mii....."Unlock all 32 Expert Staff Ghosts in Time Trials mode. Alternately, win 5,100 races."
To change your mii's driver class, go to your mii channel and change your mii's weight. That way you get to use all the different vehicles