Dishonored: Death of the Outsider

The hardest part of this achievement is getting past the arc pylon and the wall of light once you've made it into the bank and knocked everyone out.

Easiest root in is just to buy the knock out gas from the black market dealer. You can skirt around the outside of the map to get to her with very few distractions. Keep to the pipes behind the salon to avoid the dogs, foresight through the bars of the boat in the waterway and grab the painting while you are back there, and then just wait for the guard patrolling the area near the black market shop to turn his back.

I find that going through the bridge like rooms between roads is the easiest, as you play the floor is lava over to the bank. Keep to the right hand side of the bank for easiest access, watch out for the dogs as they are jerks. But by using the lamp post and that long balcony I was able to sneak over to the loft. IT WILL RUN OUT OF JUICE, DONT TURN IT ON YET. Instead, pull the whale oil out, hide near the entrance, wait for the guard to wander back and talk about his kiss (foresight is your best friend), then grab him and knock him out. I hid him behind the lift, in the garden. Watch out for the dog hanging out by the whale oil machine, you can easily hit him from that balcony you used to get over there, just wait until after you've taken care of the guard to get him otherwise it gets pretty hairy.

The guards on the roof shouldn't be an issue, they are easiest enough to spot and keep track of with foresight. Put the poison in the shoot, and head inside.

DO NOT TOUCH THE GUARDS! They need to be asleep from the Magic Juice, and knocking them out will not count. Sometimes it is just safer to bamf over them, then walk to close.

Make your way to the main level, and open the door to the hardest part from the menus behind the tellers desks.

Keep to the lamps to avoid being killed by arc pylons, they are not your friend, and then be careful of the clockwork soldiers in the next area. You can easily get to high ground as soon as you get through the door from the arc pylon room, and I recommend it.

From the far right hand side wait for the second clockwork soldier to head to the basement, then send your farsighted down to the main level, through the wall of light, up the stairs, and then by the clear windows. You can now bamf over there with your sweet moves.

Head up to the third floor to grab the info you need from the office, then bamf onto the elevator platform, open that hatch up, and then head up to the top level. Watch out for the hanging arc pylon, but do bamf over to the desk you can see on the other side. Inside the drawer is a note and a level tool. Look back towards the arc pylon, and bamf into that open hatch on the floor. Drop down onto the vault and let that thing fly. It seems silly to drop the vault, as these dudes wake up if you open a cash drawer next to them, but for some reason crashing through two levels won't do it.

Pop those bad boys open for a different achievement (011,235,813,455,891) and then grab your prize.

Once you have made it to the roof again the achievement will pop. If you want to loot, and don't want to have to worry about all the bad stuff, I would just pop out real fast, grab that crazy rare achievement (at least at the time of this writing), and then go back inside and turn everything off or reprogram. If you do want to do that, be warned that the clockwork soldiers will wake people up if you just unplug the wall of light, so don't do that unless you want to, I guess. And the lightning in the basement is really easy to avoid if you just maneuver along those open windows to the very last one. The code guy that you don't need anymore is in there, as are all of the lockers. Steal from that doctor for your third time (sadly no achievement for that), and then do as you will.

The most important thing is to take your time, and save often.

Hope that helps you out!

Other solutions don't let you know how to get this done on Original Game+ but it can be done either way. This solution will help you no matter what playthrough you are on.

Lets break down the description: "Emptied the vault without tampering with any security systems, leaving everyone unharmed and asleep."

First of all: you must leave everyone unharmed and asleep inside of the Vault. To do this, get the Poppy Tincture. Either 750 Gold from the black market or at the auction (same spot where you can drop the target through the trap door in the previous level). Take it to the roof by adding whale oil to the lift (If you are facing the entrance to the Bank at ground level, the lift is to the right and the room with whale oil is to the left). Once on the roof, insert the Poppy Tincture into the vent system and enter the Bank however you would like but the easiest at this point would be one of the two entrances from the roof. Once inside, all humans will be asleep and if you don't make any noise (run, walk or bump into a sleeping person, open cash registers, fire weapons) they will remain asleep. Make sure to leave the guards sleeping, knocking them out will void the achievement. There are a few mechanical guards from Dishonored 2 inside but these guys can only see in front of them and can easily be avoided.

Second: you must empty the vault. This is the whole purpose of the mission and can easily be done by bringing the vault to the office and looting it there or dropping it to the sewers will NOT void this achievement. For the entire time inside the bank, make saves often that you can fall back to if you mess up and not lose too much time.

Last and hardest part to figure out: Do no tamper with any security system in the bank. The buttons to open locked doors do not count against this, but you cannot disable the wall of light OR disable/retract the arc pylons OR turn off electric floors. Just displace or blink above the arc pylons using the lights to get past them.
Playing first play-through with Displace: Use Foresight to walk through the Wall of Light and set a marker on the second floor to use displace to warp up to.
Playing on Original Game+ with Blink: This is a bit trickier but should still be easy. Once you get to the room with the wall of light, you can notice there is a locked door on the second level with a window that overlooks the big room. There is a sleeping guard here that presses the controls to move the vault. That is where we need to get to. Move to the back of the room and blink on top of a speaker that is mounted on a wall, then blink onto the small ledge near that guard's window. You can then press a button to move the vault by reaching through the window AND shoot an Electric Bolt at the button to open the door to his area. Other gadgets will make noise so make sure to have one of these saved or purchased from the Black Market. That small ledge near his window is very shallow and you can fall easily from there. Once you get into the locked door you need to traverse through the electric floor area using Blink. The only tricky part with that for me was realizing you could blink on top of an open door in the hallway.

*If this solution helped you in some way, please vote positively. If you think it needs improvement, please let me know how in a private message or by leaving a comment.*

hello again, I made a guide to getting the achievement 'The Perfect Crime'

I hope I can help you.

do not forget to buy the item on the black market, toxic poppy

Its Just a Tip! The other guide is fine , I wanted to make a point that others have not mentioned!

I found that you CAN NOT unlock "The Perfect Crime" achievement in "Original Game+" run! So when you are on your "Good Old Time" Playthrough keep that in mind!
The reason is that you can not pass the Wall of Light without disabling it! I mean that wall of light near the vault and clockwork soldiers.

If you know a trick for passing the wall of light please correct me.