Here is the full list of all 12 PuppyStory achievements.
You kill 3-rd boss
You kill 2-nd boss
You kill 10 red slimes
You kill 10 strikers
You kill 10 green slimes
You kill 1-st boss
You like 10 die
You save 10 pippi
You can find 10 whistles
You kill 10 pishooters
You kill 10 spaikers
You kill 10 нуддщц slimes