Here is the full list of all 20 Twinkle Hunter achievements.
Hunt Dr. Plague
Hunt Mad Plague
Hunt Sacred Plague
Reunite with Mari
Obtain all weapons
Obtain all talismans
Obtain all fury skills
Hunt a boss with request
Hunt 6 bosses with request
Hunt a boss with Salamander Staff, Extra Heat and request
Finish the hunt with a fury skill
Hunt a boss without talisman and fury skill
Hunt one boss
Spend more than 3 minutes on the tutorial
Try hunting a boss five times
Try hunting a boss fifteen times
Try hunting a boss thirty times
Hunt a boss with 2 weapons
Hunt a boss without dash
Obtain all coins