Here is the full list of all 20 Neko Navy achievements.
Unlock all the achievements
Complete the game for the first time (any character, any difficulty, any number of continues)
Complete the game without continues (any character, any difficulty)
Complete the game on Hard or above without continues (any character)
Complete the game on Death without continues (any character)
Complete the game with Ginnosuke (any difficulty, any number of continues)
Play on Death for the first time
Unlock all the cats
Play as each playable cat at least once
Obtain more than 150 gold ingots from one Bomber
Release 10 Bombers on one stage
Obtain the 1UP item in stage 3
Obtain the ticket on stage 1 that leads to the secret route
Defeat the boss of the backside of stage 3 without taking damage in the battle
Defeat the mid-boss of stage 6 without taking any damage in the battle
Have 5 credits
Wait until the last boss self-destructs
Die by crashing into the flower of the boss on stage 2
Reset the game with the LEFT+Z+X+C command
Complete a stage without using any Bombers