Measure for the placement of your hole, and place a dot on the face of the slate tile. Cover the area where you are going to cut with three or four layers of masking tape that are large enough to cover the hole and extend outward for an inch or two. Transfer your original mark up through the tape so you have a clear starting point after you finish layering the face of the tile.
Place the tip of the core drill bit onto the mark on the slate. Apply pressure as you slowly squeeze the trigger of the drill to guide the pilot bit into the stone sufficiently to hold the core bit in place. Once the core bit itself reaches the surface, hold the drill steady to let the bit dig into the slate.
Depress the trigger on the drill completely once the core edges have dug into the surface. Apply pressure on the butt end of the drill while you push the bit into the slate installation. Go until you have cut through the slate, release the trigger and pull the core bit out of the hole with the piece of slate inside the bit. Repeat the process for as many holes as you need.