Here is the full list of all 16 Heli-Cats achievements.
Complete trial mode
Complete any mission in arcade mode with a 100% destroy rate
Obtain at least 13 additional bombs in one arcade run
Destroy all beacons in one arcade run
Reach arcade mode mission 9
Destroy arcade mode’s mission 8 boss with continues
Run out of time on arcade mode’s mission 8 boss
Destroy arcade mode’s mission X boss
Run out of time on arcade mode’s mission X boss
Destroy another mode’s secret mission X boss
Complete arcade mode without continuing and with the good ending
Complete another mode without continuing and with the secret ending
Complete arcade mode mission X with Cat A
Complete arcade mode mission X with Cat B
Complete arcade mode mission X with Cat C
Complete arcade mode mission X with Cat D