This achievement is for completing I am the Dark Night mode which is basically new game+ but with only 1 life. If you die, your save is erased and you have to start over. This may seem difficult but in reality it is not that difficult to achieve.
#1 - keep an eye on your health. If you hit the red, press start immediately and hit restart. You may lose some progress depending when the last checkpoint was but it is 100% better than starting over
#2 - This achievement ONLY counts the main story. So ignore everything else. There is no need to risk your save any of the side missions or assassins.
#3 MAKE SURE TO DEAL WITH ALL SNIPERS BEFORE DOING ANYTHING. The Snipers are the most annoying enemies. Avoid the ones in the Diamond district but there will be times where you have to go against some. They are priority #1.
#4 In Predator sections do not take risks. Be smart and take advantage of your gadgets. First and for most take out Snipers first. You can easily thin the herd by using your Sonic batarang and Remote claw. Sonic Batarang gives you a free knockout and the Remote claw can take out guys using propane tanks and you can even us it to attach enemies to vantage points. Take advantage of your environment as well. Detonate those mines to take out henchmen as well as explosive gel on weak walls.
#5 In Combat Scenarios take advantage of your combos. Combos will net you quick kills. If you are good with counters without icons you will be fine. I find it is easiest to knock any gun users down, eliminate the giant thugs and armoured guys then focus on the rest. Martial art guys can be a pain but an easy way to deal with them is to get a combo, knock them down and use the multi ground takedown. Take advantage of the knife users for an easy takedown by dodging.
#6 Bosses - They are fairly simple. Just follow the pattern
Croc - Stun then attack and when he picks up the canisters mash the quick batarang to explode them. There is noting special about him.
Deathstroke - He can be a bit tough. His fight relies on countering. If you get into a fight cutscene(not sure what else to call it), watch for the slowdown, that is the signal to counter however the attack I find doesnt have one. Here my advice is to chain a couple of hits together and use a takedown on him. Time your attacks as he will attempt to counter. You can also catch him with the batclaw slam although dont over use it as he will catch on.
Copperhead - She is a bit annoying but not all that difficult. Spam the quick explosive gel to deal with all her doubles, when the real one is exposed just attack relentlessly. Rinse and repeat. Eventually she will start dash attacking so be sure to keep dodging to avoid being hit by that.
Bane - I found this fight to be the hardest. First half isnt too difficult. Ultra stun and attack, use takedown to deactivate his venom or whatever it is then attack and use a takedown or two again. After you go outside, its a bit harder. Its the same sequence but he charges you and slams the ground and enemies join the fight. Honestly ignore the henchmen. Run and let him charge. Dodge as he slams and go for the ultra stun + takedown. Then continue using take downs and repeating the pattern until he has 1 health bar. He will do 1 more charge sequence before a cutscene.
Firefly - Not a difficult fight do the same thing you have always done. Hit him with the glue grenade, batarang spam and then bat claw. Be sure to spam the counter button when you are grapelled to him and flying around to make sure you counter.
Bane #2 - First fight is the same as the 1st. After that he will Hulk out. You want to sneak up and silent take down and throw him into the electric cells. After doing that twice, jammer will activate. Hide until the coast is clear and disable it then sneak up and repeat.
REMEMBER: RESTART ANY TIME YOUR HEALTH GOES INTO THE READ DURING A BOSS FIGHT
That is all there is too it.