BioShock Infinite: The Complete Edition

I don't know if this is a proper solution, per say, but it is a collection of tips and hints to help you clear sections of the game. (copied from my XB360 version of this achievement)

I should also preface this by saying this guide assumes you have already beaten Burial at Sea Episode II at least once, and assumes you're not using this for a single playthrough to get all of the achievements. It's meant for players only seeking this achievement and not all the others.

Stealth, as you already know going into this, is key. There are things you can do to minimize your chances of being caught.

1) If enemies are in the area, remain crouched while moving unless you have to move quickly. You are generally more silent and will remain hidden behind small cover.

2) Certain actions will give away your position and draw enemies to your attention, so be aware of these three things:

A) Walking through any puddle or stream of water, even while crouched. Walk around them if you can.

B) Walking over glass shards, even while crouched. They will crunch loudly. Walk around them if you can.

C) Hanging onto any Air Grabber hook for too long. It takes a while, but you'll hear them buckle and start to snap. You will not fall, but your enemies will hear this.

Now, these can be good things, especially if you are trying to lure them around the corner so you can use your melee knockout, or get them to move out of potential cover to hit them with your crossbow without leaving cover yourself. But you should be aware of the three things regardless.

3) Unless they are in your way, there is little to no benefit in actually engaging enemies in combat. Avoid combat if you can, particularly in the Columbia level and after the Return to Rapture ambush.

If you must go into combat, a few things to remember.

A) Fire and Ice splicers are resistant to the effects of both the Tranquilizer Bolts and the Gas Bolts. Either Bolt will stun them but they will recover unless you headshot them. In all of my playthroughs, I had to follow up either Bolt with a melee attack to actually knock them out if I didn't hit them in the head.

B) Helmeted enemies cannot be subdued with melee. Use Possession or the crossbow. Hooded enemies, on the other hand, can be subdued with the melee attack.

C) The AI is notoriously bad at simply coming around corners when they are in the "yellow" alerted stage. Use this to your advantage, especially once you've upgraded Peeping Tom so that you can turn invisible and remain in place without wasting Eve. You can take out multiple enemies this way if you get spotted in Columbia.

D) You can often recover your Tranquilizer Bolts from downed enemies. If you are forced to use them and can search the sleeping body, do so.

4) Do not rely on Possessed Turrets for offense. Because the achievement is based on using non-lethal methods, Possessing the turret will prevent them from attacking you but they will not attack enemies for you.

5) Do not waste time getting Audio Diaries, Voxophones, Secondary Objectives, or Upgrades to Old Ironsides unless you want to. None of them are required or will really aid you.

6) Don't bother exploring while in Columbia. There is nothing there you need unless you are low on supplies. Get in, get out. There are plenty of alternate paths available by using vents or tunnels under the floorboards. After you remove the security lockdown, it might be easier to go invisible and make a mad dash to the right back into the room with the Songbird Prototype and just backtrack your way through the lab. You will encounter single enemies or pairs of enemies on your way, but nothing like the hoard of enemies in the stairways outside the room with the lockdown switch.

7) The best place to ride out the Return to Rapture ambush in Suchong's lab is the locked door with the Turret and Old Man Winter Upgrade. There is only one way in and you can hide around the corner. This provides you ample cover and forces them to attack you from a single vantage point. How you deal with them is up to you. I melee'd when I could, and used Tranquilizer Bolts when I had to, but using Possession or Old Man Winter are also good options. You will face approximately 2 Fire splicers and 6-7 regular enemies, so don't leave the room until Elizabeth taunts Ryan about having won the fight.

After you have survived this ambush, go outside the Silver Fin restaurant and go invisible. Ryan and Atlas's thugs are going to fight it out for a while. Eventually Ryan's forces should win this, but the combat will reduce the total number of enemies on the field. MOST of them will congregate on the stairs in front of the Manta Ray lounge after the fight. You can sneak by them using Peeping Tom if you want, or use Gas Bolts to knock most or all of them out.

Engage as few people as you can in the Lounge after that, but don't bother sparing any of your remaining ammunition or health packs. Your only objective is getting to the elevator. You will not have any further combat once you get into the elevator, so Tranquilize and Melee whomever you need to, use noisemakers for distractions, use Peeping Tom to run while invisible, and burn all of your health packs if you take damage.

8) Lastly, it is possible to just be brazen and run and gun the entire game (minus the Big Daddy, of course) if you are quick with the crossbow and adept at ducking behind corners and going invisible. My last playthrough of this I was significantly less stealthy than my previous attempts and just went after the enemies. The key to this method is to not get overwhelmed and always loot the unconscious enemies to recover your bolts.