Step 1: Remove the old lock from the door. Place the new lock against the door and use a pencil to mark where the opening needs to be enlarged. 

Also check if the recess for the faceplate (the metal strip on the side) needs to be wider or longer.

Step 2: Use a drill with a wood drill bit to enlarge the existing mortise. Drill overlapping holes in the wood to achieve the desired width or depth.

Step 3: Use a sharp chisel and hammer to remove the remaining pieces of wood between the drilled holes.

Step 4: If the new faceplate is larger, use a chisel to adjust the recess on the side of the door so that the plate sits flush with the wood.

Step 5: Slide the new lock into the mortise to check if it fits well and is flush with the door.

Step 6: Check if the holes for the door handle and the bathroom lock are still suitable. If necessary, re-drill these holes with an appropriate wood drill bit.

Since you have chosen door hardware with a backplate, you will directly cover up old screw holes and drill holes neatly.