This story mission is the continuation of The Atonement and will automatically start as soon as you get on the plane. You’ll also be able to call in Nick as back up. It’s worth having him to make it easier to shoot down John’s plane. Tight dogfights isn’t necessary easy so make sure to gain some good distance so you can effectively fire your guns at him. Your plane also has unguided rockets but they can be quite hard to aim and hit.
After successfully shooting down John’s plane (you don’t need to destroy it, just deal enough damage), he’ll be forced to land in the woods and will attempt to escape. Follow the mission marker conveniently tagged at him and pepper him with bullets and rockets until he’s disabled. After that, there will be another cutscene before he dies. Mission complete.