Play endless mode on a map of your choice with settlements ON, ideally, play on easy too if you don't need any distractions.
Set up your base and try and get as many scouts up and going ASAP! From here explore the Frostland and eventually, you'll be tasked with making contact with settlements one by one. I found my first after the first storm and the second and third after the second storm.
Once you've made contact they'll start giving you choices, generally, the top one will be BAD
and gain you distrust; the second option will be GOOD
but will gain their trust! You'll want all three to be loyal to you at once. This can be challenging as I initially had lots of arguments over cores where you will gain rep with one and lose it with another settlement. They should settle down after a while especially after you've fully upgraded their camps.
I got all three loyal by day 100 exactly! It's worth noting that my achievement popped after about a minute or two whilst reports across forums is that it may not unlock immediately.
For reference the relations levels go;Distrustful - Cautious - Neutral - Friendly - Loyal
By the time you've likely got all three loyal to you, you should have a well-established camp. I'd recommend from here to keep playing and when you get options from camps to then start choosing the BAD
options to start working towards the Distrust achievement (Make a save after, just in case!).