Here is the full list of all 32 Bleed achievements.
Defeat the Laser Bots with an S-Rank.
Reflect a bullet that was reflected by the Rival.
Defeat the Mantis Core with an S-Rank.
Defeat the Chopper Core with an S-Rank.
Defeat the Spider Core with an S-Rank.
Defeat Gibby & Stu with an S-Rank.
Defeat Battalion with an S-Rank.
Defeat the Mirror Core with an S-Rank.
Defeat the Dragon Heart with an S-Rank.
Defeat the Bubble Core with an S-Rank.
Defeat the Rival with an S-Rank.
Investigate the pit in Guppy's Manor.
Win a Challenge Mode fight against three opponents.
Recite some poetry atop the Hall of Heroes.
Clear Story Mode on Hard (or higher).
Clear Story Mode on V. Hard.
Clear Story Mode on Normal (or higher).
Defeat White Mk II with an S-Rank.
Defeat Guppy with an S-Rank.
Clear Story Mode on Easy (or higher).
Defeat Bunny Rockette with an S-Rank.
Clear Arcade Mode on Easy (or higher).
Clear Arcade Mode on Normal (or higher).
Clear Arcade Mode on Hard (or higher).
Clear Arcade Mode on V. Hard.