Use the suicide/desperation attack (CIRCLE attack) more than 30 times. The needed limit is less on earlier stages than later stages, but use the attack this many times to be sure.
Stage 1 at least 15 times
Stage 2 at least 15 times
Stage 3 at least 20 times
Stage 4 at least 20 times
Stage 5 at least 30 times
Stage 6 at least 30 times

Clear all the enemies from each room or hallway you come across until no more enemies respawn. The enemy count also includes the rooms you re-enter by accident.

Enter "bloody mode" at lteast eight times in a given stage. Some stages have a higher requisite limit than others.
Stage 1 at least 4 times
Stage 2 at least 6 times
Stage 3 at least 8 times
Stage 4 at least 8 times
Stage 5 at least 4 times
Stage 6 at least 4 times

Perform a 25-hit "just frame" combination in a stage (it may have to connect with an enemy to count). Notice sometimes the sword has a wavy effect when attacking -- you pressed the SQUARE key at just the right frame to do extra damage (or 'Cool Attack'). The timing of the button press is delayed slightly on hits 3, 6, and 9. The combination statistic must be at maximum for this challenge to be possible.

Get a continuous set of combo hits/kills in a preset period. The period differs depending on the stage.
Stage 1 about 13 seconds
Stage 2 about 15 seconds
Stage 3 about 17 seconds
Stage 4 about 19 seconds
Stage 5 about 21 seconds
Stage 6 about 23 seconds

Kill the boss of the stage under a preset period. This varies depending on the stage. The timer only starts when you first face the boss.
Stage 1 under 20 seconds
Stage 2 under 30 seconds
Stage 3 under 60 seconds
Stage 4 under 35 seconds
Stage 5 under 40 seconds
Stage 6 under 45 seconds

Clear a stage without being struck by an enemy or enemy attack. Note O-Ane-Chan may still lose life from "bloody mode" or the desperation attack and fulfill this challenge.

Clear a stage while still in "bloody mode". Very easy if you anticipate fighting the boss.

Clear a stage with the lifebar flashing in warning. Very difficult if you do not have the invincibility gem against some bosses. Planning ahead helps.

Clear a stage with a 100% bloody sword. Very easy if you couple it with "bloody mode".