So I got this achievement quite by chance this evening.
From what I can tell as other solutions state, explosive damage will outright kill instead of causing a soldier to bleed out. The same goes for Viper Binds the Mutons' execute ability.
I had my Grenadier positioned by a car on a Retaliation misson (My very first as it happened) and she was attacked by a faceless - He probably took 3/4 of her health, and caused the car to explode but that did not cause extra damage, and she went into the 'Bleeding Out' stage.
Once a soldier begins bleeding out you'll have 4 turns to evac him/her. You MUST evac the soldier, simply completing the mission will not pop the achievement, you must select one of your soldiers, press RS to select an evac zone and have the soldier carried and evaced - This popped the achievement for me immediately.
I've heard that the soldiers chance of bleeding out depends on their will as well as how they are injured - Additionally you can increase the chance of a soldier bleeding out by purchasing the 'Stay With Me' perk in the Tactics School.