The story is told in the third person, and it centers singularly on the grandmother. She's the character we're told the most about, by far. She's also the only character whose point of view we can access directly. We get to hear her thoughts and feelings, although we never get too much detail. We are usually given a direct, short summary that leaves a lot of room for imagination on the part of the reader. (That there is room for interpretation with regard to the grandmother's inner thoughts continues to be a subject of debate).
The only other character who is given comparable attention is The Misfit. Interestingly, we only learn about him through the grandmother's perspective. After the grandmother is killed, though, there is a brief switch to The Misfit's perspective (although we don't get any further into his head). Some early critics really didn't like this shift in perspective at the end. They claim that O'Connor's handling of point-of-view is sloppy. On the other hand, if she wanted the story to continue after its main character died, what choice did she have? Besides, the story might be about the beginnings of The Misfit's transformation as well.