Here is the full list of all 20 Twinkle Hunter achievements.
Hunt a boss with request
Hunt one boss
Try hunting a boss five times
Hunt Dr. Plague
Hunt Mad Plague
Hunt Sacred Plague
Reunite with Mari
Obtain all weapons
Obtain all talismans
Obtain all fury skills
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
Spend more than 3 minutes on the tutorial
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