There are 3 hunters (per class) and 3 monsters in the base game.
You start with 1 hunter of each class as well as 1 monster (the Goliath.)
To unlock the second character of each class (and second monster), you earn 1 Star with each piece of equipment of the first character.
To unlock the third character of each class (and third monster), you do the same thing with the second character.
Any challenge I didn't list below (including all Monster Level 1 Masteries) simply require you to deal damage. This damage can be done to anything, opponents or creatures.
These challenges don't have to be done in multiplayer; you can do them in a solo lobby.
Maggie Level 1 Mastery -> Unlocks Griffin
Pet Trapjaw - You must follow Daisy for 2600 meters. This is pretty much only viable in the Hunt game mode.
Harpoon Traps - Harpoon the monster 40 times.
Griffin Level 1 Mastery -> Unlocks Abe
Sound Spikes - Plant 15 sound spikes at least 50 meters apart.
Harpoon Gun - Harpoon the monster for 1 second 14 times.
Val Level 1 Mastery -> Unlocks Lazarus
Medgun - Heal 16000 damage.
Tranquilizer Gun - Tranq the monster 14 different times.
AP Sniper Rifle - Place 12 targets on the monster.
Lazarus Level 1 Mastery -> Unlocks Caira
Silenced Sniper Rifle - Place 200 targets on the monster.
Lazarus Device - Use it 4 times.
Personal Cloak - Cloak revive 7 creatures. - Can be regular revive or Lazarus Device.
- Note - it says creatures. Animals do count towards this (Lazarus Device only)
Hank Level 1 Mastery -> Unlocks Bucket
Shield Projector - Shield teammates from damage 20 times.
Bucket Level 1 Mastery -> Unlocks Cabot
UAV - Cover 3000 meter in Hunt or Rescue.
Edit - Some users in the past reported this unlocking on startup. This may have just been some quirk of the free weekend they had running on the game back then. I'm going to assume anyone new to the game now actually has to unlock the hunters until someone posts otherwise.