Acrid
Complete the …To Be Left Alone challenge; finish the Void Fields (Hidden Realm found by activating the Null Portal in the Bazaar Between Time).
Under the Bazaar is a hidden cave going to a purple portal to the Void Fields. Beat the nine challenge cells in the Void Fields to unlock Acrid.
Artificer
Complete the Pause. challenge; free the survivor suspended in time for 10 Lunar Coins.
Collect 11 Lunar Coins, then find a Newt Altar. Pay one coin so the blue portal spawns upon charging the teleporter to 100%. Use the blue portal and head to the shop. Pay 10 coins to unlock Artificer.
Captain
Complete the Washed Away challenge; beat the game.
Commando
Unlocked by default.
Engineer
Complete the Engineering Perfection challenge; complete 30 stages. Can be done in multiple runs.
Huntress
Complete the Warrior challenge; complete the 3rd teleporter event without dying.
Loader
Complete the Guidance Offline challenge; defeat the Alloy Worship Unit in Siren’s Call.
Will spawn upon destroying five vulture nests in Siren’s Call in the large open area of the map.
Mercenary
Complete the True Respite challenge; obliterate yourself at the Obelisk.
MUL-T
Complete the Verified challenge; complete the first teleport event 5 times.
REX
Complete the Power Plant challenge; carry the Fuel Array from the (back of the) Escape Pod to the Abyssal Depths.
In the Abyssal Depths, locate the broken REX and interact with it to unlock REX.
!! MUL-T and Acrid are unable to obtain the Fuel Array solo, since they spawn without a pod.