I'm trying to use 4e's skill challenges in Pathfinder, but I have a question: should I ask my player to roll for initiative and take turns to Win the challenge, or just the best PCs should play it?
As others have said, it's mostly personal preference. One of the problems with 4e's take on skill challenges is that it doesn't always give equal screen time to each PC.
In the early drafts of 13th Age and in some of the 13th Age organized play documents, they have a concept called the "montage" in which, during travel scenes, investigation scenes, or exploration scenes players describe a challenge they faced and a way they were able to get past the challenge. It takes place around the table.
I'd probably combine these two ideas and go around the table. It's faster than rolling initiative and gives each player some screen-time. Each result can lead to the next scene and so on. This also helps ensure you "fail forward" and not have a bad roll lead to a simple failure but instead to an interesting if difficult resulting situation.
Depends on the level of involvement and whether it makes sense in the fiction they would be going in order.
For instance social skill challenges might not make sense since not everyone can meaningfully contribute. A physical skill challenge that is required due to impending doom? That might make more sense to keep organized.
Having played a number of 4e Skill Challenges under Living Forgotten Realms, in many different ways, I can say I've seen different DMs do both.
Some DMs just let whoever is best at a task go for it, particularly when the challenge at hand only has a narrow set of skills available.
Other DMs don't want any one player to hog the spotlight, especially when the challenge allows a wide array of skills, and so will just go around the table one at a time, rather than roll Initiative.