HITMAN 2

So this achievement differs from the original release a bit.

As of now, there are only 2 escalation contracts.
The Videl Cataclysm and The Corky Commotion.

EDIT: According to z 4assed monkey, The Corky Commotion does not count towards this achievement.

I did the Videl Cataclysm. Here's the strats:
I would recommend getting mastery up to the point where you can start in the Attic.
Level 1 - Kill Christophe Nightlinglam as a security guard.
Start in the Attic. Equip pistol, fiber wire and coin.
From the start, turn around and head down the ladder and pacify the guard. Take his uniform.
Go to the target and fiber wire. Pacify the tech crew guy if you want. He may or may not get you in trouble if you leave him alone.
Exit.

Level 2 - Kill Christophe Nightlinglam as a security guard; you can only wear the security guard for 60 seconds.
Same as level 1.
Only difference is after you kill the target, change into the target's clothes. I would recommend also taking out the tech crew guy in the room too.

Level 3 - Kill Christophe Nightlinglam as a security guard; you can only wear the security guard for 60 seconds; kill Fabrice Sansouci as a security guard.
This one is a little tricky.
From the start, drop down the ladder and drag the body to the door you head out of towards the targets. Move the body as close as possible to the door without showing line of sight to the guard near the door in the next room.
Now grab the letter opener in the room.
When you're ready, change into the security guard uniform and book it to the first target.
He will be sitting down in a chair by the time you get to him. Equip the letter opener and adjust yourself so you aren't in any line of sight of anyone nearby. Throw it at the target and book it to the next.
From there, it's the same as usual. Fiber wire, change into the target's clothes, pacify the tech crey guy, exit.

Level 4 - Kill Christophe Nightlinglam as a security guard; you can only wear the security guard for 60 seconds; kill Fabrice Sansouci as a security guard; some things have been added/removed.
The only addition to level 4 is that there is a guard in Christophe's room which makes fiber wiring the target more difficult as well as a guard in the same hallway as the first target who has direct line of sight.
In planning, replace either the coin or the fiber wire with a remote explosive or breaching charge (I only had access to the remove explosive).
Again do the same as you did before, with the dragging and grabbing the letter opener.
After you kill the first target, run away as fast as you can because the guard will be looking.
For the second target, stand next to him and drop the remote explosive. Run away.
After you're a safe distance away, detonate. Then exit.

Level 5 - Kill Eric Hebert as auction staff; kill Alain Rocheleau as auction staff; you can only wear the auction staff disguise for 60 seconds; some things have been added/removed.
As you can see, they threw a curveball and changed the objective for level 5.
Because of this, change your starting position to the AV Crew start. Keep the pistol, remote explosive and coin as your loadout.
From the start, run to the target Eric Hebert first. He will go upstairs if you don't take him out first and it'll be annoying to get him back down.
Run into the room with only one auction staff member inside (big room that's practically empty with the food and such).
As Eric is passing by the room, throw a coin to get him inside. It may take 2.
Afterwards, pacify him. DON'T kill him yet.
Then, close all the doors in the room (there's several). Pacify the other auction staff guy.
Now drop the remote explosive near Eric.
Change into the auction staff disguise from either person and then run towards the other target.
Once you're a safe distance away, detonate the explosive.
As you get to the second target, dispose of him however you want. He will be alone.
Once you kill him, change into the target's clothes to cancel out the timer.
Exit and you're done with the contract.

Feel free to provide feedback!

Found this easiest to do on The Teague Temptation. This was on the XB1 version, in case there's any variation between platforms.

Start in the AV area with a silenced pistol, a lockpick and coins. Hide your pistol (and later, the letter opener) when not in use.

Level 1+2:
Immediately run to your right towards Courtland Aupry. Throw a coin near the equipment between you and him, so that the witness near him is the one to investigate. As he's bending down to pick up the coin, you want to push Aupry over the railing. Immediately run back the way you came and through the door, then down the stairs on your right, all the way to the kitchen. Head around to the right and through the exit. Because you won't pass any cameras, this path can be taken for Levels 1 and 2 identically.

Level 3+4+5:
Same as before: Immediately run to your right towards Courtland Aupry. Throw a coin near the equipment between you and him, so that the witness near him (closer to you) is the one to investigate. As he's bending down to pick up the coin, you want to push Aupry over the railing.

Now, run back to your starting point, wait for Benoit Lagarde to leave for the bathroom, and shoot the camera ahead of you when he's out of earshot. Run up and to the right, and at the other end of the area is another security camera. Take cover behind the crate and shoot this camera too. Run forward and to the left, and on your right is a door you can unlock. This is the Sheikh's suite, if you're fast enough he might still be out on the balcony. For now, take cover behind the long desk in front of you.

Wait for the Sheikh to walk inside, he'll take a phone call. Then he'll walk back outside onto the balcony. There's a security guard in here, you need to distract him to grab the letter opener on a desk on your right. So work your way up to between the two couches, and shoot the wall at the far end of the room, so the guard looks that way. Quickly run up and grab the letter opener, and exit the way you entered. Run all the way back to where you shot the first camera, and head through the doorway to the bathroom on your left. Here you'll wait for Lagarde.

Make sure your letter opener is hidden when he walks in, then sneak up behind him, equip the letter opener, and silently kill him. No need to hide the body, just start heading down to the same kitchen exit as before. Because you're not pacifying anyone else, and not changing disguises, this will work the same for the final three levels.