I'll start it off: The Hunter: Go through an entire level only using a hunting rifle.
The Alone Wanderer: Kill every person you come in contact with.
Raiding Party!: Using Jericho, kill an entire town and loot their junk.
Mayor: "Found" a town (using like, ruins or an empty place or junk) and uphold the law there.
Badass Sniper: Kill an enemy without them noticing you using only a sniper from several hundred feet away.
I Am Legend: Without light, with Dogmeat, and only using one type of gun/weapon, go through a metro tunnel or any other ghoul/super mutant infested area and kill as many as you can.
Tough Skin and Strong Boned: Without apparel or weaponry, kill every enemy in an area/quest.