Trim drywall away from the top of the door jamb with a drywall saw to make room to run the cable above the door.
You may have to remove some baseboard to be able to grab the wire from along side the doorway, or if you want to continue to run the wire behind the baseboard.
Mark the wall at the top of the baseboard to give a reference for cutting once you have removed it.
Score the top edge of the baseboard. This will release the paint and caulk so you don’t peel the paint or the drywall paper when you remove it.
Use a broad knife and a screwdriver to gently pry the baseboard from the wall . Take care not to mark the wall above the baseboard.
Use a drywall saw to cut the drywall along the floor line. The tip of the saw can be used to poke a hole in the wall.
Drill the holes for the cable high enough to avoid nails in the bottom and middle of the stud so the cable will be far from possible nail intrusion.
Drop a weighted string or sash chain down through the gap between the jamb and stud on the side of the doorway until it can be seen at the bottom of the trim piece.