You Have Been Rylakinated!

Here's how you get this achievement:

1. 
   
2. First of all, make sure you are on the garrison campaign quest Garrison Campaign: Darktide Roost(A) / Garrison Campaign: Darktide Roost(H).
3. 
   
4. Complete the quest line until Dirty Rats (complete it as well). Remember to loot the goblins as you need [item=116978] to interact with the vehicle.
5. 
   
6. Kill more goblins untill [item=116978] drops.
7. 
   
8. You need to control a [npc=86085].
   Be aware that the achievement tracker instantly starts as you enter the vehicle. So make sure you got everything under control before you jump in!
9. 
   
10. Kill 10x [npc=85357] within 3 minutes. The injured ones and the whelps do NOT count towards the achievement. I suggest starting at the west side of the island and moving clockwise.
11. 
    
12. Profit!
13. 
    

You do not have to be on the daily quest (Assault on Darktide Roost) to complete the achievement. You can complete this achievement at any time, assuming you did the garrison campaign quest first.

Edit: Fixed information.

As part of the Darktide quest chain/daily, you will have a quest called Dirty Rats to destroy 5 [npc=86085]. While you are doing so, goblins will drop an item called [item=116978]. Hold on to these as they will allow you to drive the shredders later on.
After turning in the quest (the turn-in is south of the shredders), you can go back and enter the shredders to do this achievement.

There are several shredders by the coast; I picked the one in the camp at 57.41, 88.12 which was closer to the actual spawns of the [npc=85357]. It "costs" 1x [item=116978] to enter a shredder, so if you fail the first time, you can farm another item and try again. To prepare for the achievement I went through and marked out a path using raid markers to make sure I had 10 [npc=85357] available. Next, I cleared other mobs to make sure aggro wasn't a problem.

Note: You and the shredder do not share health. You actually have separate health pools and can die before your shredder does!

When you are actually IN the shredder, you have 3 abilities:
#1: [spell=176060]
#2: [spell=172324]
#3: [spell=176105]

It seemed to work well running up to a Rylak, channeling a flame thrower on them, then spamming #2 until they die. If I got any other mobs on me, I spammed #3 and they died pretty easily.
As mentioned in another comment, the timer for the achievement starts immediately after you enter the shredder. Start running and killing right away!

I was not able to test if these are shared credit with another party member, nor test whether killing other people's tags counted for credit.

Good luck; happy hunting!

Here's a Tomtom waypoint list for those of you trying to figure out a path. The Rylakinator closest to the start of this path is at 56.08, 89.93. I've included it in the list to make it easier.

/way 64.63, 89.09
/way 63.29, 85.31
/way 61.95, 83.02
/way 61.08, 83.29
/way 60.48, 83.25
/way 60.08, 84.44
/way 59.24, 84.97
/way 58.28, 84.79
/way 57.96, 86.05
/way 57.36, 86.97
/way 57.18, 88.86
/way 56.55, 88.95
/way 56.08, 89.93

I counted 11 Darktide Adolescents in all, and the last one on the path is pretty far away. I haven't actually attempted this yet myself, as there are too many people currently in Darktide Roost. I'll update with my success whenever that comes.

EDIT: I attempted it and can say that it works very well, I also found another Adolecsent and have added it to the waypoints. I finished with almost a minute to spare.