Oh man, this is my absolute favorite 'what if' scenario to hunt down. I'm always digging through AO3 tags like 'Akatsuki Naruto' or 'Naruto Joins Akatsuki'—you'd be surprised how many there are, and they go in wildly different directions. Some are just pure crack where he's Pain's weird little brother figure and they order terrible pizza, but the good ones really dig into the psychology.
There's one I remember, 'Deviant' by Laluzi on FFN, that handles it fantastically. It's not a quick heel-turn; Naruto ends up with them after a genuinely tragic divergence post-Wave Mission. The slow corruption of his idealism while he's trying to 'fix' the organization from within is heartbreaking. It's less about him being evil and more about him being utterly, tragically lost. That specific story got me hooked on the trope because it treated the premise with real weight.
The tone can swing from grimdark tragedy to surprisingly wholesome found-family stuff. I've seen a few where he bonds with Kisame over shared outsider status, or drives Konan up the wall with his messiness. The dynamic with Itachi is always a huge draw—that mentor/student, older brother/younger brother tension is a goldmine for angst or unexpected camaraderie.