Balloonicorn pets are pretty scarce these days, making this achievement rather difficult to obtain. Here are some tips to help you in your quest:
1. ALWAYS equip the Pyrovision goggles regardless of your class or loadout. This will allow you to see who has Balloonicorns equipped.
2. Once you have found an opponent with a Balloonicorn, relentlessly target them and try to avoid switching sides. On the other hand, if one of your teammates has a Balloonicorn equipped, look for the first opportunity to switch sides so you can target them.
3. Consider sending a friend request to players with Balloonicorns and ask them if they would be willing to boost. This will speed up the process considerably.
4. Troll idle servers searching for players with Balloonicorns. Once you have found one, you can target them repeatedly to rack up the kills. (NOTE: This is how I obtained the achievement.)