This achievement can only be won in the game mode Fighter Squadron. It is awarded for killing the opposite side's heros, i.e. the Rebel Alliance's Millenium Falcon and the Empire's Slave I.
As battles end as soon as one side has collected 200 points, this gives you very limited time to make this work. In addition, it is almost impossible to kill the heros with ordinary fighter craft, unless you team up with various teammates. However, attacking with non-hero fighters makes you vulnerable to attacks from enemy fighters, to whom you act as an easy target with your full attention focused on the hero.
Thus, this solution will focus on fights with you commanding a hero ship yourself.
As soon as you start up the game, fly close to the ground at low speed and circle the area. After a while a hero icon pickup will spawn somewhere in your vicinity (it looks like a circle with a bar in the middle - or a lightsaber with a glowing aura). Note that the spawn points are random, but they are generally somewhere in the middle of the map.
Collect the hero pickup, then press
to transform into the Millenium Falcon (if playing as Rebel) or Slave I (if playing as Imperial). Make sure to not fly into anything or get killed while collecting the pickup, as you lose it when dying.
Now continue with the battle until an opponent has also become a hero; you will notice "[Gamertag] has become Millenium Falcon" (respective Slave I) in your battle log. Try to fly high so you get a good view of the battlefield; identify the opposite hero and chase him down. Do not waste time attacking anyone else.
As soon as you are chasing the opposing hero and reach firing range, push back
to divert your full energy into your weapons and thus increase damage dealt. You will notice that this slows you down as well, which makes sharp turns easier to accomplish.
Keep firing, but be aware of your weapons overheating; include quick pauses of 1-2 seconds to avoid a longer forced pause while your overheated weapons cool off. Use your torpedoes as soon as they become available (
for the Falcon, both
for Slave I).
Both ships are so strong that you can almost ignore incoming fire from other fighters. As the Falcon, activate your shields when available (with
) to add more protection; as Slave I, try to use the sensor jammer (
) as much as possible so that other fighters can't get a lock on you.
Feel free to use evasive maneuvers (
) to avoid incoming missiles, but use this only rarely to not lose precious attack time.
Keep at it, and eventually you will destroy the hero. In most cases it will be a close call to the end of the match, also depending on the skills of your opponent. Note that the kill will only count towards this achievement if you are the one to give the opponent the final blow. It will show in the battle log as "[Gamertag] has destroyed Slave I/Millenium Falcon"; your achievement counter in the achievement app will also advance to 50% respective 100%.Update:
After the March 2016 update, it is no longer possible to progress with this achievement by simply ramming (and thus destroying) the opposing hero:"Fighter Squadron: Vehicles of equal size will now both be destroyed when colliding regardless if one vehicle is shielded"