No matter how good you are, you're not going to hit Beiber Fever levels. Not that that's a bad thing. But if you publish a book or two, you might actually gain a level of fame as a published poet.
That feat might even be harder to accomplish than becoming a major pop star, though. There are those super poets like Anne Carson and Nikki Giovanni who have millions of fans. However, it's more likely that the pinnacle of your success will be the solid respect and recognition of students and your academic peers in a university setting.
Which is fine though, right? We know you weren't thinking of getting into poetry for the money, so you probably didn't expect to become incredibly famous, either.
There are certainly famous people who write poetry, but it doesn't exactly get called poetry. Lyricists—you might know them better as songwriters—actually come up with the poetry (or lyrics) a song uses. If you've ever heard an Elton John song like Goodbye Yellow Brick Road, you're listening to the poetry of Bernie Taupin.
That makes poets of artists like Bob Dylan, Bruce Springsteen, and Madonna, and rappers like Kanye West and Nicki Minaj—and they're about as famous as it gets, although they all have some kind of music behind them. If you're going for the full-fledged music-less variety, we wish you the best of luck. Tell us when your book is coming out, we'll totally tell our friends.