This is a story-related achievement. Once you beat the Plant-boss the chapter finishes and you get the achievement.
This fight has three tstages. Every stage consists of two parts.
1. You have to decrease the life-bar of the Plant-Boss itself.
2. The Plant-boss will have mutations. These mutations are: orange membrames, tentacles and red neck. Every instance of these mutations will have its own mini-lifebar.
Advise on weapons to use.
Plant-boss can be shot with any weapon you have available.
But my preferred weapon is the flail-gun. One shot of the flail-gun will take out a third of the Plant-boss. Another advantage is the wide reticle on the weapon. You have to do some real effort to miss the boss with this weapon. :-)
Mutations can also be shot with any weapon you have available.
But my preferred weapon is the leash. One grab-shot, not the thumper, will take out one instance of the mutations. This works for the membrames and the tentacles. For the red neck I again used the flail-gun. The leash has some advantages: uses up no ammo, auto-targeting.
While fighting this boss you want to constantly strafe. This will make it harder for the Plant-boss to grab you. However once you get grabbed there is no need to panic. Quickly tap the red B-button on your controller and you're free to fire another shot.
If you prefer to use the PMC then you might run out of bullets. Luckily there are some ammo-packs in the arean. I believe there are three packs. Just make sure you don't pick these up when you're full of ammo.