In terms of a first person, second person, third person point of view, "My Last Duchess" is told from the first person perspective. This means that the speaker or narrator is also a character. This poem is a monologue spoken by the Duke to an unknown listener who does not verbally respond in any way. Readers are alerted to the first person point of view in the first two lines when we see the usage of "my" and "I."
That’s my last Duchess painted on the wall,
Looking as if she were alive. I call
That piece a wonder, now;
It's not always true, but generally, the person narrating a story to readers from the first person perspective is somebody that winds up being our hero character and trustworthy. I believe that readers are likely to start this poem with that assumption, and that is what makes the monologue so great. By the end of the poem, readers realize that the Duke has a completely warped view of himself, women, and the world around him. We, and the silent listener, should absolutely not trust the Duke.