Here is the full list of all 33 Mitsurugi Kamui Hikae achievements. It takes around 8-10 hours to unlock all of the achievements on Windows.
Defeat stage 2 boss "Kaiser" (any difficulty)
Defeat stage 3 boss "Suzuka" (any difficulty)
Defeat stage 4 boss "Magatsu Kaina" (any difficulty)
Defeat stage 5 boss "Magatsu Suzuka" (any difficulty)
Use Zanshin to defeat 100 enemies
Watch the tutorial
Defeat stage 1 boss "Kaina" (any difficulty)
Defeat 10 enemies
Defeat 100 enemies
Use Zanshin to defeat an enemy
Use Zanshin to defeat 50 enemies
String 50 hits
String 100 hits
Perform the holy technique "Celestial Judgement"
Perform the holy technique "Divine Wind"
Perform a Just Guard
Perform 10 Just Guards
Reach "Mitsurugi" zeal level
Clear Easy difficulty
Clear Normal difficulty
Clear Hard difficulty
Clear Very Hard difficulty
Defeat stage 6 boss "Magatsu Hino Tsurugi" (any difficulty)
Defeat 1000 enemies
Perform a Counter
Perform 10 Counters
Perform a Flourish
Perform 10 Flourishes
Complete all upgrades
Collect 100,000 SP
Max out Life Gauge
Max out Katana Gauge
Clear Inferno difficulty