Here is the full list of all 20 Samurai Kento achievements.
Open 1 treasure chest.
Open 5 treasure chests.
Open 15 treasure chests.
Open 25 treasure chests.
Get 25 money.
Get 50 money.
Get 100 money.
Get 200 money.
Reach Level 2.
Reach Level 5.
Reach Level 10.
Reach Level 20.
Defeat 25 enemies.
Defeat 50 enemies.
Defeat 100 enemies.
Defeat 200 enemies.
Defeat the First Boss.
Defeat the Second Boss.
Defeat the Third Boss.
Defeat the Fourth Boss.