No. His highest rank is Genin. However, this is only a rank and he has the power of a kage. He also mixed his wind element affinity into the resengan and even took it further by creating Wind Style: Rasenshuriken, something that was deemed impossible to do. Even the 4th Hokage couldn't do it. So, he actually surpassed Kakashi and mastered some kage level jutsu.
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In Naruto, the way a Chunin becomes a Jonin is by being selected by other Jonins and maybe taking a test. They haven't really explained that process as much.
He needs to become a chunin first. He may just go straight from genin to Hokage tho too. As far as the series goes now tho, he is just a genin who is the stronger then any ANBU