If you ever wondered how to do it, you probably have checked out already the achievement page ta TrueAchievements website, but the solutions there are for 2-players boosting, where one player has to disconnect and re-connect and bla bla bla.
I will tell you how to do it solo and it is not that hard.
First of all, the frame to use. Stealth is the essence of my play-style so I went with Ivara. Standard mods nothing weird - mine is built for duration and efficiency for long lasting prowling.
Do not bring Kubrows or Kavats as they may go and kill someone. Do not put weapon on the sentinel, as then the sentinel may kill someone, and DO NOT cast any spectres. I tested this, if your spectre kills someone, it is added to your kill count and makes you fail with this achievement attempt.
You do not need a sniper, but if you are going to attempt head-shot achievement you need something powerful and precise, I suppose you can pick anything you have, I went with Vectis Prime.
As a secondary - this is where the trick is - you need something that does a little damage, and has high status chance. I went with Staticor just because it has built in radiation damage and no other damage types at all. I modded used following mods on my Staticor
+60% Electricity and +60% Status chance
- Perpetual Agony
+30% Status duration
+60% Heat and +60% Status chance
- Stunning Speed
+40% Reload speed and +10% Status chance
Make the weapon completely silent
- Sure Shot
As you see, I only used 6 mods and left 2 slots unused. This setup changed its stats as follows:
- Noise from Alarming to Silent
- Reload from 1.5 to 1.1
- Status chance from 25% to 61%
- Radiation damage from 19 to 42
Before you run to attempt the mission, go to the Gear setup and equip the Extinguished dragon key, You will want to reduce the amount of damage you will do with your Staticor.
There is only one place where this achievement can be obtained - it is Void, Oxomoco. When mission starts, put down energy pad and get your energy maximised, or if you use Energy Siphon mod / arcane on your warframe just wait till you have plenty of energy for Prowling.
Put the prowl on, and GO!
Do the first shot/kill with your sniper and after done check the stats if it was counted as head-shot. If not, quit and start again.
Now you want enemies do all the killing and you just spectate. put away the sniper and from now on, only use the under-powered Staticor. Stay invisible and shoot one of the enemies. if he gets the "radiation icon" then his team mates turn against him and they start fighting each other. Rinse and repeat till end of the game. Take your time and double check who you are shooting before hitting the trigger. If you shoot enemy who is really low at health, your Staticor shot may kill you and mess up your achievement attempt.
If you are running low on energy, find a "quiet room" and deactivate prowl, pop an energy pad and get your energy reserves up again.
If you shoot enemy in the room with turrets, even the turrets start shooting that enemy and usually killing it quite quickly.
As now the game keeps spawning enemies even after X out of X enemies are killed, you do not have to kill everyone, so you can leave the ones with lets say less than 10% health/shields behind and don't worry about them. Only shoot the enemies you know will not be one shot kills with your Staticor.
As I said, take your time, but don't take too long time, eventually Lotus will tell you, "mission failure is eminent" and then you have 60 seconds to finish and extract.