You get this achievement after successfully escaping the Tortured Path chapters without anyone having purchased
any Blitz/perks. So if someone else buys a perk nobody will get the achievement.
For the Tortured Path you get a damage bonus corresponding to your tortured path level up to a +200% Damage Bonus at Rank 50.
It works in private by scaling to the total rank of everyone in the group. However, it is still much less of a damage bonus than public. (From Firstmate Fizz)
Therefore its best to go for this in Public, and with a group of higher Ranked people you'll find its not really difficult.
This doesn’t need to be done in one go so if you complete map 1 & 2 but fail map 3. Then you’ll only need to do map 3 at any other time.
Besides that the only other tips are the basic:
-With all the extra money you'll have from not buying perks, definitely keep buying Starter Weapons for 500 (as its the cheapest) until everyone has Jack-In-the-Boxes as they come in handy.
-Have an Upgraded 9mm SAP as its the best weapon for Bosses, Wüstlings, etc.
-Make sure to have someone standing in the extraction zone in the end to ensure you complete the map (suggestion from Seitzz)
Now playing through the chapters in public with the damage bonus is definitely the easiest way to do this but as a side note it turns out that if you complete the Easter Egg to get all the perks given to you, it doesn't negate the achievement. However, I’d advise doing this only for map 3 as it’s the easiest of the Easter eggs.
For info on how to do the EE refer to this guide:
So perhaps if you're unlucky like me and get disconnected on wave 8 of Chapter 3 or just fail maybe you dont want to have to go through chapters 1&2 again to get to 3 in public.
So you can choose chapter 3 in custom, do the ee steps without buying any perks, and by wave 5 or whenever you finish everyone will get all the perks for completing the EE. Since it’s custom there won’t be a damage boost but You can now continue and escape the map having all perks and still get the achievement since you technically never purchased any yourself.
If you have any suggestions let me know.