Dead Man Walking doesn't fit well into a classic plot analysis—it doesn't have one plot so much as two plot swings, one involving Pat and one involving Robert. But Shmoop will not be deterred. If you have to bash the book into five bits, here's how it would go.
Exposition (Initial Situation)
Prejean tells us about her background, her commitment to social justice, and her somewhat casual decision to be a pen pal for a Death Row inmate. This is basically Chapter One—a short set up.
Rising Action (Conflict, Complication)
Prejean becomes more involved with Pat's case and tries to get his death sentence converted. No luck: despite her great efforts, he is executed, anyway.
Climax (Crisis, Turning Point)
Prejean decides she will never, ever, ever be a spiritual advisor to a Death Row inmate again. Naturally, she changes her mind, and that's the turning point that sets up the rest of the action.
Prejean becomes Robert Lee Willie's advisor and sticks with him until his execution. She also works with Vernon and Elizabeth Harvey, the parents of Robert's victim, Faith.
Yeah, it's a little odd to think of this as falling action, since Robert's execution is probably the emotional center of the book. But the build-up to Robert's execution in many ways recapitulates the build up to Pat's. The action is falling not so much in the sense that tension is diminishing as it is diminishing in the sense that you've seen this same tension before.
Prejean explains her ongoing work against the death penalty and with victims of criminal violence. It's not a very neat ending, but hey, this is a nonfiction book, and let's face it: reality doesn't have a lot of neat endings.