Here is the full list of all 21 Burst Fighter achievements.
Clear 1 level
Clear 100 levels
Destroy 1000 enemies
Destroy 100000 enemies
Finish Campaign in Casual Mode
Finish Campaign in Normal Mode
Finish Campaign in Expert Mode
Destroy 3 enemy bosses using Blade wing and weapon
Finish Campaign in a single run
Finish Campaign in a single run and never die (Normal or Expert mode)
Reach 69000000 in total score or more
Die in the first 10 seconds of level
Defeat KR4K3N
Defeat M3DUS4
Defeat 4R4CHN3
Defeat 4RGU5
Defeat C3RB3RU5
Defeat CH1M3R4
Defeat H4D35
Defeat L1CH
Defeat M0TH3R