Previously this was host only but apparently after a recent patch this is now obtainable by everyone playing. (Thanks to Aura Battler for this info). You'll get this naturally if you're going for the other achievements. Below is a list of levels that unlock characters in case someone needs it. To unlock the chefs you only need to finish the level with at least one star.
The chefs aren't named and the human chefs are difficult to describe in text, so for photo reference, visit here:
https://dotesports.com/the-op/news/overcooked-2-chefs-guide-...World 1-2: Human chef
World 1-5: Cat chef
World 2-1: Human chef
World 2-4: Human chef
World 3-1: Human chef
World 3-3: Raccoon chef
World 3-5: Crocodile chef
World 4-2: Octopus chef
World 4-4: Human chef
World 4-6: Human chef
World 5-2: Eagle chef
World 5-5: Cat chef
World 6-2: Squirrel
World 6-5: Robot
Kevin 1: Human chef
(Unlocked in world 1-3 by getting a combo of 2)
Kevin 3: Human chef
(Unlocked in world 2-4 by getting a score of 600 for solo or 1000 for co-op)
Kevin 5 : Beaver chef
(Unlocked in world 4-3 by getting a combo of 4 for solo or 5 for co-op)
Kevin 7: Alien chef
(Unlocked in world 5-5 by getting a combo of 5 for solo or 8 for co-op)