Fit an electric drill with a diamond-tipped tile drill bit. Put an 'x' of masking tape over the spot on the backsplash where you want to drill.
Press the drill bit onto the middle of the 'x' and then press the trigger on the drill. Apply heavy, steady pressure on the drill until you feel it go through the glass and into the backer board.
Lighten the pressure as the drill enters the backer board. Go through the backer board or drywall and then slowly pull the drill out of the hole. Use an can of pressurized air to blow any debris out of the hole.