The drywall can be cut below the wall cabinets and removed and replaced with drywall or cement board (for tile installation. This can be done without removing the cabinets, and you can cover the seam with molding, or run your backsplsh up to the cabinets. If you can't salvage the drywall, this is actually a better approach than repairing hacked up drywall. By installing cement board you will get a flawless installation surface for the new tile.