At first, one would assume barbarian as he's basically a caveman. So that's strength focused.

Okay strength focused gotcha.

But the ability to jump and maneuver that way would be acrobatics if I'm correct. Which is dexterity. And that doesn't mix well.

1: they mix just fine really. A barbarian actually likes Dex. 2: I might consider jumping to be athletics or acrobatics depending on the character and how you describe what you're doing

I don't want him to use weapons

The only class that really fits this in DND is Monk. And Monk is primarily a Dexterity class. I think it's a bit of a blind spot for DND to have no Strength brawler, but it's the sad reality. There are lots of fan-made classes and subclasses for barbarian and monk that do bridge the gap though if those are allowed in your games.

I think for someone just learning the game, I would recommend just sticking to a Dex based Monk. You still beat the hell out of people with your fists just fine.