While no means as annoying as Choose Your Path, this achievement can still be quite annoying simply for the rarity of the targets. To that end, before you start conciously grinding this make sure you upgrade your Fighter class ship. I had level 3 unlocked on damage increase, offensive power increase and engine power.
Early in any given match you won't be presented with many opportunities to add to your tally because your opponents will need to earn the battlepoints required to use a hero ship. If you're playing offence this is probably a good time to recklessly play the objective, and if you're on defence then maybe casually not protecting your objective is your best bet. You want the match to push into the second and third stage if possible, as this will give your opponents every chance to play a hero ship.
Periodically check the scoreboard
to see if anyone on the opposite team has the triangular star logo next to their name which indicates they're in a hero ship then do your best to take them down. Sometimes hero ships can be difficult to discern from regular ships (Luke/Poe if it's an X-Wing match, for example), so try to look out for the player's name to make sure you're focused on the right player. If anyone manages to kill you while in a hero ship the game will helpfully mark them in orange for revenge once you've respawned. Some heros can be a challenge, I often found Darth Maul hard to kill because of his stealth ability, and Luke/Poe have crazy efficient healing abilities.
You'll often have competition from other players on your team (a few hero ships present big targets) but if you focus on just trying to take down heroes then you're likely to take down one (maybe two) a match.
Also: When I was looking into this achievement originally there was talk that assists, either by the hero crashing or being killed by an ally, counted towards the total, but from my own observations the tracker only moved when I personally defeated the hero ship.
If anyone has any evidence to the contrary, let me know in the comments!
EDIT: It would appear to be confirmed by multiple sources that assists do not progress this achievement. Also that Darth Maul sucks...
SEVERAL UPDATES LATER EDIT: As per skhter
in the comments: "Might want to add that this is slightly easier now with the added hero ships mode. You can suicide at the start and instantly spawn into a fighter jet. Then you'll have 3-4 enemy hero ships to hunt."