BioShock Remastered

This achievement essentially means Play The Game on Hard WITHOUT dying. Vita-Chambers are only used to re-spawn after death so not using them means don't die. This means you need to continually save your progress. If you die you must load up an earlier save.

There is a simple method for ensuring that you never use the Vita-Chambers. Before Starting a New Game, go to Gameplay Options, Scroll Down to Disable Vita-Chambers and make sure it says On. This ensures ALL Vita- Chambers are disabled in the game, this is really important and means every time you die you load a previous save. I highly recommend doing this.
Also this achievement is for HARD difficulty only; there is a higher difficulty called Survivor but this unlocks on HARD only.

Two Methods to Obtain this. A legitimate and an Exploit.

THE EXPLOIT
Start a new game on easy. Again either disable the Vita-Chambers or ensure that you never die. Reach the final boss. As you are taking out the final needle of Adam press pause, and change the difficulty to Hard. Then complete the rest of the scene as normal and the achievement will unlock.

Legitimate Method
With Vita chambers disabled your new best friend will be the Save option. Make sure that you save every few minutes, particularly at the start of the game when you lack plasmids and your health is low. etc. Later in the game you will be much stronger but still save often, its quicker to save than replaying sections of the game.

Check all crates, trash cans, dead splicers, etc for extra first-aid and eve items. Turn on item Glimmer in the options, you will be able to spot loot much easier. Also increase the brightness a touch this will allow you to survey your surroundings.

Most guides advise you to stick solely to mission objectives, however I believe I benefited from scouring the environment. My ammo was always full and I always had reserve health/eve packs.
Buy Health/Eve Upgrades and Tonic Slots ASAP
Upgrade your weapons wisely. Leave the pistol until last
Purchase/ collect as many Tonics as possible.. Unlock all Tonic Slots as soon as you can. Engineering tonics are a massive help for hacking and getting reduced prices. Physical/Combat tonics can make this playthrough a doddle. Make sure you equip the scrounger perk

Buy Target Dummy ASAP. Ignore all other Plasmids and Plasmid Upgrades until later in the game. (Dont waste ADAM on Plasmid Upgrades etc, Plasmids like Insect Swarm/Cyclone Trap are pretty useless anyway. As long as you have Incinerate, Electro Bolt, Telekinesis + Target Dummy you will be fine.) You will notice that as you progress in the game the upgrades that were once costing a fortune are now majorly discounted. Eg Electro Bolt 2 was 120 Adam in Neptunes Bounty by the time I reached Fort Frolic is was 20.

Make sure you photograph each enemy and gain the max research bonus, this unlocks unique tonics but also gives great damage bonuses. The auto turret hack is a great help and the Big Daddy damage bonuses turn a 5 minute battle into a 2minute battle.

Rescue every Little Sister in the Game, not only will you get the ADAM but after every 3 you rescue you get a bonus of 200 Adam and after every 6 you recieve a unique Tonic.
Save Electric Buck and Grenades for Big Daddy battles. Upgrading your shotgun and Grenade launcher are a massive help

On a sidenote people have reported the achievement not unlocking, this was an issue on the 360version and occurs because all of the game completion achievements are unlocking, all you need to do if this happens is load up the final boss fight again and defeat it, with all of the other achievements already unlocked their is nothing to obstruct this achievement.

play on easy too you stick the last needle in fontaine and change too hard remember too disable vita chambers before u start a new game, but i did it on easy up too i stuck fontaine with the last needle so it works

Just an addition to the previous solution. The game has to be completed (or glitched) necessarily on hard, can't be on Survivor.
Also, even with patch the glitch still works (june 23,2018)

it's been almost 7 years, yet here I am to share something I couldn't see anyone else mention.

You can obtain this achievement in your New Game +.
So basically if you where like me and completed the game but missed some achievements such as this one, you can start a new game plus and get this achievement even faster.
As already mentioned, start the game on hard, change it to easy, and then just put it on hard at the end of the game, and remember to disable the Vita Chambers from the beginning to avoid ruining this achievement.

Also as a side note if you happen to not get the achievements for investigations / weapons this is your chance to do it from the beginning and with your camera and weapons upgraded from where you left them...
Ggs, and I know it's late lol.

This Achievement can also be done in NG+ so what I did was 100% it on Survivor than replay it on NG+, it is very easy this way and you can get through it very quickly.