This Achievement is easily gained, whether you are playing with a friend or on your own.
When playing with a friend, both characters must be linked to separate controllers while you are still in the Batcave/Arkham Asylum. Start and finish any mission (Story or Freeplay) with your friend. Assuming you are both signed in to different Gamer Tags, you will both get the Achievement.
If you are playing on your own, you can mimic the above scenario by turning on a second controller (or any peripheral with a D-pad and face buttons) and making sure that both characters are linked to separate controllers before starting the level (again, in Story or Freeplay). You can have one Gamer Tag signed in, or sign in as another one if you feel it is needed.
As long as both characters are being controlled at the beginning through to the end of any level, you will get this Achievement.