Old ceramic tile that has been mortared to the wall will need to first be broken and then pried off. Wear gloves and safety glasses -- these pieces are sharp. Use a hammer and tap the center of each tile. Depending on the age of the tile, they may crack, or you may need to use more force. Break each one away and place the pieces in a small garbage can as you go.
You will have exposed the old mortar or tile mastic that is on the wall. If it is very old, you can chip it away with the end of a prybar or a 4-inch solid drywall knife. Start in one corner and begin chipping it away. This is a laborious process. You will probably find that some pieces come away easily, while other pieces require more strength and chipping to remove. Keep working until the surface is completely clear of mortar.
If the mortar is stubborn, you will need to use an adhesive remover. Mineral spirits may work or you may need a stronger chemical. Wear a mask during this process and open all the windows in the room. Turn on a fan for better ventilation. Apply the remover, let it set for a few minutes and then begin scraping away the mortar with your drywall knife. Again, it helps to have a small trash can near by to help keep the area clean.
Once all the mortar is removed, the surface may still be uneven, or there may be small, stubborn bits of mortar left. Use a heavy grade of sandpaper, a block sander works well, or if necessary, an electric hand sander, and completely sand the entire surface. It should be smooth to the touch when you are done. This will give you a completely clean slate so that you can replace the backsplash.