In the battlemode, you can hire an additional 4th dragon from a friend. Cheapest dragons cost 2 energy bars extra, so a total of 3 energy bars for one battle. After one battle you either have to wait for the energy to refill or have a friend gift a refill to you. Repeat this 9 more times for the achievement. Since you have an extra dragon, I advise to use it for completing higher battles or 3 starring them. Do keep in mind you need to win the battles fot the achievement. You don't actually have to use the dragon from your friend in-battle, you just need to hire him and win the battle you hire him for, even if you succeed in doing so with just your first 3 dragons, it still counts.
Note: you can see your progress, by selecting the "trophy symbol", while in your dragon village. Not the green one, but the other one.