So your game basically consists of fighting a long series of bosses, right? How do you make a game story out of that? How do you ramp up the drama other than simply ramping up the difficulty?
Answer - You start asking...Why?
Why do these bosses need to die?
And in answering 'why', you get your story!
This kill is to save the village.
This kill is for mercy.
This kill is to save the country.
This kill is to answer a betrayal.
This kill is to save the world.
After that, it doesn't matter what storytelling technology you have at your disposal, whether it be cut-scenes, NPC dialogue or flavour text. You'll be able to get that clear and simple message across somehow.
You must fight this Boss because...
Once the player has their reason firmly in mind, their actions will tell the rest of the story.