Basically, complete the game. Escape is strange; I was able to reach the escape and trigger the ending while ignoring the objectives. I'm unsure if it counts/unlocks the achievement.

• A Fresh Start - Completed the Tutorial
  
• Smells Fishy - Completed the Shipping Yard
  
• Cashing Out - Completed the Casino
  
• FORE! - Completed the Country Club
  
• Living in Luxury - Completed the Mansion
  
• A Fine Art - Completed the Art Gallery
  
• Trail of Chaos - Completed the Escape

Objectives. Take note for Mansion; it is possible to grab the wine (triggering alarm) without shutting the generator. There is a secret bookshelf in the cellar that leads to the back where the generator is. Unsure if Art Gallery can be completed without locating the map first but it's beneficial to do so anyway.

• Key Item - Locate the key and use it to open the locked crate in the Shipping Yard
  
• Mid-Heist Theft - Slip the key from a Guard's pocket to the safe in the Casino
  
• Punch it in - Locate the codes to access the secure lockup and steal the documents in the Country Club
  
• X Marks the Spot - Locate the map and steal the painting in the Art Gallery
  
• The Finest Drop - Shut off the generator and steal the prized bottle of wine without setting off the alarm in the Mansion

Out in the Open
Complete a level without using hiding spots (excluding Tutorial)

A one-life or no-checkpoint run. Hiding spots are the green items we have to manually interact with using "E".

Like the Wind
Complete a level remaining completely unseen and unheard by all guards (excluding Tutorial)

Unheard refer to the player's footsteps so go light on the sprints. Getting partially spotted (guards investigating with ???) counts as being seen. Tools are allowed (including distraction tools) so make use of them. Stompers are most likely consider footsteps so avoid that.

Skilled Thief, Master Thief, Creature of Chaos
I did these 3 together in a single run as it made sense. For Master Thief, having coins, cherry bombs equipped & unused didn't give me the achievement so I replaced them with other tools. For Creature of Chaos, all guards on the level needs to be alerted (with an exclamation on their heads).

• Skilled Thief - Complete your objectives and leave the level without using any tools (Tutorial not included)
  
• Master Thief - Complete a level without using distraction-based tools (No Coins, Cherry Bombs or Stompers.)
  
• Creature of Chaos - Draw the attention of every guard in a level (Tutorial and Escape excluded)

One Tool runs
Use only one tool. The description can be misunderstood as "allowing" Socks and Stompers but they are instead disallowed. Depending on the approach, pick the best tool for the job. Refer to my video runs for the tools used on each mission.

• Shipping with One Tool - Completed the Shipping Yard using only one tool (Excluding passive tools, Socks & Stompers)
  
• Casino with One Tool - Completed the Casino using only one tool (Excluding passive tools, Socks & Stompers)
  
• Country Club with One Tool - Completed the Country Club using only one tool (Excluding passive tools, Socks & Stompers)
  
• Mansion with One Tool - Completed the Mansion using only one tool (Excluding passive tools, Socks & Stompers)
  
• Art Gallery with One Tool - Completed the Art Gallery using only one tool (Excluding passive tools, Socks & Stompers)
  
• Escape with One Tool - Completed the Escape using only one tool (Excluding passive tools, Socks & Stompers)

Undetected
This refers to no guards having an exclamation mark on their heads (alerted), investigations (??? on their heads) caused by being partially spotted or from footsteps are allowed.

• Ghost the Shipping Yard - Remain completely undetected throughout the Shipping Yard
  
• Ghost the Casino - Remain completely undetected throughout the Casino
  
• Ghost the Country Club - Remain completely undetected throughout the Country Club
  
• Ghost the Art Gallery - Remain completely undetected throughout the Art Gallery
  
• Ghost the Mansion - Remain completely undetected throughout the Mansion

Hot Pursuit
Evade capture by a group of 3 guards or more

This should mean having at least 3 guards in pursuit before shaking them off. The smoke pipe tool works great here; get 3 guards on our tail and smoke and run. I did this on Art Gallery (video below).

What was that noise?
Distract a guard with a cherry bomb attached to another guard

Art Gallery has a quick setup at the start. Enter the office and plant the cherry bomb on the static guard and use it (video below).

Footsteps as distraction
Can be used to lure guards away from their post or break to their rhythm without spending a tool. An advanced use is to repeat it in the dark (on guards without flashlights) to lure them far away from their original spots. Can be used for Ghost runs but not "Like the Wind".

Forcing an investigation by getting partially seen
By walking out into the "risky" light setting, we can force an investigation to our location. This can be used as a opening move to lure guards to darkness before continuing with footsteps. Do this with ambient lighting and not the guard's flashlights. Can be used for Ghost runs but not "Like the Wind".

Breaking a pursuit
Once alerted, guards will be able to track us even in the complete darkness. Breaking their line of sight will help stop them, either by a sharp turn or with obstacles. Useful for Creature of Chaos.

Mission 0 - Tutorial

Mission 1 - Shipping Yard
Mission 2 - Casino
Mission 3 - Country Club
Mission 4 - Art Gallery
Mission 5 - Mansion
Mission 6 - Escape

Like The Wind - Unseen & Unheard
Skilled Thief, Master Thief, Creature of Chaos - No Tool & All Guard Alerted

What Was That Noise
Hot Pursuit