Easy difficulty:
Lose twice on Normal.

Hard difficulty:
Beat the game on Normal.

Critical (Issen) mode:
Beat the game on Hard.

Oni Hard mode:
Beat the game on Critical (Issen) mode.

Genma Puzzle game:
Solve all the puzzles in the game.

Secret Training mode game:
Beat normal training mode and finish the game.

Oni Shooting game:
Beat the game on Normal.

Ultimate mode:
Beat the Oni Shooting game.

Adventures of Heihachi mode:
Beat the game on any difficulty to play as Heihachi.

Original Onimusha swords:
Beat the game and acquire all three items from the Phantom Realms. Now start a new game to play with all three swords from the original Onimusha.

Bishamon sword:
Beat the game with Onimusha rank, or pick it up in the Phantom Realm in Honnouji.

Ako's Black Haori:
Beat Issen/Oni training.

Michelle's alternate costume:
Beat the game on Normal.

Michelle's second alternate costume:
Beat the game with Onimusha rank, or play the game with a memory card inserted containing a saved game from Onimusha Blade Warriors.

Jacques' alternate costume:
Beat the game on Normal with Jacques and get a high score, or beat the game with Samanosuke's alternate costume.

Samanosuke's alternate costume:
Beat the game on Normal with Samanosuke and get a high score, or beat the game with Jacques' alternate costume.

Panda costume:
Beat Secret Training mode, or play the game with a memory card inserted containing a saved game from Onimusha Blade Warriors.

Alternate ending:
Max out all seven Haori of Ako.

Thanks to Revolution readers G-sauce and tubbs!