This is pretty easy to do in a public quick match. Choose normal for longer time.
Poison can be acquired a few ways: from bunkers, air drops or crafting.
The easiest way to do it without relying on other people and RNG supply drops is to craft them at the cooking station in the cabin.
You need 1 Jerry can and 1 herb/root.
I did all 10 in one public match by first making a sickle, 1 rock and 1 wooden log.
This makes it easier to collect berries and herbs/roots.
Then start to collect 6 berries and 3 herbs/roots. You can make 3 jerry cans per 6 berries and 3 poison from the 3 jerry and 3 herbs.
Now to poison. You can poison terminals that open bunkers, each terminal counts towards progress. You can also poison traitor crates as well as survivor crates, supply drops, objectives, dead bodies, dead animals and objectives.
This method will get you it quick and within 30 mins whether you are survivor or traitor and provided you aren't killed by traitors.