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