Here is the full list of all 25 Ikaruga achievements.
Destroy Final Chapter Boss: Tageri.
Destroy Chapter 4 Boss: Misago.
Destroy Chapter 3 Boss: Uzura.
Complete Chapter 3 in any mode with a grade of A or better.
Destroy Chapter 2 Boss: Buppousou.
Complete Chapter 2 in any mode with a grade of A or better.
Complete Chapter 2 in any mode with a grade of S or better.
Complete Chapter 1 in any mode with a grade of A or better.
Complete Chapter 1 in any mode with a grade of S or better.
Complete Chapter 1 in any mode with a grade of S++.
Destroy Chapter 1 Boss: Eboshidori.
Complete Chapter 4 in any mode with a grade of A or better.
Complete the Final Chapter in any mode with a grade of A or better.
Complete Chapter 3 in any mode with a grade of S or better.
Complete Chapter 4 in any mode with a grade of S or better.
Complete the Final Chapter in any mode with a grade of S or better.
Complete Chapter 2 in any mode with a grade of S++.
Complete Chapter 3 in any mode with a grade of S++.
Complete Chapter 4 in any mode with a grade of S++.
Complete the Final Chapter in any mode with a grade of S++.
Clear the game in any mode without using Continue.
Clear a certain chapter in any mode without shooting at all.
Complete one chapter in any Co-op mode without using Continue.
Complete three chapters in any Co-op mode without using Continue.
Clear the game in any Co-op mode without using Continue.