In a game I'm in, we are in a village that has their own traditional dance. The bard wants to learn the dance to get an "in" with the villagers, would a Performance check allow him to copy the moves if he saw/studied in an hour or say?

Sure a Dexterity or Charisma  (Performance) check could do it.

Basically, whatever check the DM thinks would be appropriate for a bard learning a dance or song.

Bards are presumably trained and spend a lot of their lives picking up songs/dance/stories from the people they interact with. So I would probably go with Performance (charisma) as a roll to see how well they pick it up. No failure but the higher the number, the better the job they do. DC 15 or 20 would be sufficient to show that they learned it very well, maybe even impressive. Even a DC10 would be a good showing for such a short time. DC5 would be a flawed performance showing the bard didn't really grasp the elements.

However, if the bard in question doesn't have proficiency in performance (maybe it isn't something they specialized in) .. you could go with Deception (charisma) to see how well they fake it. Alternatively, an Int check might be appropriate if the emphasis is on learning but not really performing - though even then a non proficient performance (charisma) check might still be the best choice.

I'd let anyone do this, because all these average townsfolk can. The check would determine how long it would take.

The rules in Xanathar's have Intelligence reduce the amount of time it takes to learn something.  If you want to randomize how long it takes to learn a dance, the way most consistent with the RAW in Xanathar's would be to have it be an Intelligence check, possibly Intelligence (Performance).

Now there's a combination you don't see too often.