Who in the story Me Before You is the "me"?

That is difficult to say because it is never explicitly stated in the novel or the film; however, the author has addressed it to an extent. Intending to be cryptic, the author left it without explanation in order for it to be able to represent both of the characters, to an extent.
The idea is that the book is exploring the lives of Will and Lou before and after they meet each other and how they are radically changed by interacting with one another. In the story, Will's perception of life is changed by the joy Lou brings him, and Lou's world is opened up by Will's adventurous spirit and his worldly experience. The title is meant to represent each characters' past ("Me") before they met the other ("Before You").