The bosses in Inkwell Isle 1 are as follows:
Root Pack (Botanic Panic)
Goopy Le Grande (Ruse of an Ooze)
Hilda Berg (Threatenin' Zeppelin)
Ribby and Croaks (Clip Joint Calamity)
Cagney Carnation (Floral Fury)
Beating the Run n' Guns aren't necessary for the this achievement, but you might as well go for them to get some coins to buy new weapons and some achievements later on.
Here's a guide for finding and defeating all bosses, the guide is also a A+ Walkthrough, so it might be useful for other achievements: