To snatch up this achievement, you'll need to take down the following eight assassin's all out for the 50 million dollar bounty placed on the Bat's head:
1. The vicious Killer Croc - Story related, you'll fight him in the introductory sequence.
2. The Shock Glove armed Electrocutioner - Story related, you will 'battle' him early on upon boarding Penguin's cruiser.
3. The veteran mercenary Deathstroke - Story related. You can also go for the 'One Eye Open' achievement while you're at it.
4. The Spanish super-soldier Bane - Story related, you'll fight him frequently throughout the campaign.
5. The mysterious Copperhead - Story related, you'll fight her after apprehending Black Mask
6. The unstable pyromaniac Firefly - Story related, you'll fight him briefly on the bridge after disarming the bombs.
7. The unrivaled marksman Deadshot - Side mission. Alfred will contact you during the game about a S.O.S signal towards the upper left hand side of Gotham. Heading there will trigger a side mission, which will entail you doing some detective work before taking down Deadshot in the Gotham Bank.
8. The unmatched Lady Shiva - Side mission. As you progress throughout the game, keep an ear out for a crying baby. Use detective vision to look for the source of the sound should you have trouble finding it (orange audio waves will emit from the origin point). Once you've found it, Shiva will appear, and you'll initiate her side quests.
During your final face off with her in Wonder City, you can attempt to go for the 'Flawless Display' achievement. Heads up, restarting the checkpoint does not avoid the achievement should you take damage.
Anyway, this one's not too hard to get if you like the game and are willing to see everything it has to offer.