This one is a bit of a grind, my method was while playing team objective, just stand never an ammo crate an just continually fire either at the enemy or just into the ground/air. Wherever you fancy really, just keep doing that until you reach 10,000. Just make sure you are using the shortbow.
This took me about 20-25 hours, didn't time it exactly, but thats how much my steam played time increased.