if it was my house and my contracted job, I would remove the old tile for a better surface to work with.
If you do not remove the old tile, can you stand the extra width and thickness of the tile and the backsplash? How will it look all stacked up? How well will the tile adhere to the corners and fronts?
So, removal of the tile should not be that difficult. Sometimes a wood chisel and a light tap with a hammer will "snap" the tiles off.
When you tile your counter, be sure and use a good water proof mastic and grout. When the grout has completely cured, three days after install, be sure and apply liberally some silicon grout sealer. It will be an investment in time and money well taken.