Taking Care of Business

This is soloable on normal 10 player (did not try heroic or 25) with a warrior in Dragonflight.

1. 
   
2. Pull boss' pet and walk it around the room to each 4 corners while hugging the wall.
3. 
   
4. Kill pet.
5. 
   
6. Repeat for 2 other pets.
7. 
   
8. Kill boss.
9. 
   
10. Achievement pops.
11. 
    

NOTE: You can track the achievement to see if you walked the pets properly before killing them.

This one shouldn't pose too much of a problem. Grab each puppy as they spawn and walk them to the nearest wall, then walk them in a lap around the room. Ensure you slow DPS if one of them is going too slowly. A monk tank on the adds would help as Provoke increases their movement speed, but I did it as a Blood DK so any class is fine.

The only things of note is that the Ripped Soul will be coming from an inconsistent location due to all the kiting in P2, so have the boss in the middle so healers have a chance to top it up. In P3, move Hecutis intermittently so your healers don't get overwhelmed by the raid damage that comes from moving him. If you need to wait for a minute to let them catch up it's fine to do so.

The in-game tracker wasn't working for me but Blizzard has us covered on this one as when you complete one of the adds' laps, they will drop a pile of poop.
Barghast Poop
Hecutis Poop

Once you have 3 piles of poop you can kill the boss.

FrostyShot#1667 if you have any questions about the achievements! (US Servers)

This one is super simple, just take each add to the all corners of the room. We marked all of them to make it easier to see, and to not cut corners.

After the add "does its business", it's safe to kill it, be sure to hold dps a bit to ensure you have time to get the add around the room as the Huntsman and the dogs share health.

Hecutis shouldn't be moved quickly unless the healers can handle it.

Video guide: https://youtu.be/zklmQ0atbF0