This is one of the hardest achievements to get, not because of difficulty but because its just random and annoying.
You can get this achievement playing normally, getting 20 settlers on Sanctuary, and getting food, water, beds and stores for everyone. *Everyone must have jobs to get happiness to a high level*.
Eventually you will get 80% happiness, from there, just wait till it gets to 100%.
If you only want this achievement and you want to get it quick:
Go to Red Rocket, send one settler there, give him enough food in the workshop. And give him a store and a bed. Then AFK there, sleep, wait, and repeat until you have 100% happiness.