Here is the full list of all 12 ASTRO: The Beginning achievements.
Destroy 30 enemy ships
Destroy 100 enemy ships
Beat the game as Astro
Beat the game as Amer Din
Beat the game as Palych
Beat the game as Ling
Destroy 3 different bosses
Destroy all the bosses
Beat the first level undamaged
Die more than 10 times
Pick up more than 15 weapon capsules
Pick up more than 30 HP capsules