How can I identify uppercase and lowercase characters in a string with swift?

You could always see if the lowercase representation is different from the current value;

let string = "The quick BroWn fOX jumpS Over tHe lazY DOg"
var output = ""

for chr in string {
    var str = String(chr)
    if str.lowercaseString != str {
        output += str
    }
}
print(output)

>>> TBWOXSOHYDO

In Swift 5, we can now check for character properties per Unicode standard.

For your question, chr.isUppercase and chr.isLowercase is the answer.

By extending String and Character I think I've arrived at a fairly flexible solution that is fully(?) Unicode aware. The syntax below is for Swift 3.0. Though there is nothing here that should not be possible in Swift 2.x.

extension String {
    func isUppercased(at: Index) -> Bool {
        let range = at..<self.index(after: at)
        return self.rangeOfCharacter(from: .uppercaseLetters, options: [], range: range) != nil
    }
}

extension Character {
    var isUppercase: Bool {
      let str = String(self)
      return str.isUppercased(at: str.startIndex)
    }
}

let str = "AaÀàΓγ!2😂🇺🇸"
let uppercase = str.characters.filter({ $0.isUppercase }) // ["A", "À", "Γ"]
for char in str.characters {
    "\(char): \(char.isUppercase)"
}

// A: true
// a: false
// À: true
// à: false
// Γ: true
// γ: false
// !: false
// 2: false
// 😂: false
// 🇺🇸: false

The TODO for these extensions is to refactor isUppercase on Character to not convert to String.

I am not sure what you mean by trying to avoid C functions. I hope this does not include avoiding the frameworks and foundational libraries that OS X and iOS offer you when developing an app, like, for instance, the NSCharacterSet class which provides exactly what you need in a Unicode compatible implementation.

To expand on Matt's answer, bringing it a little closer to your question's requirements:

import UIKit

let testString = "Åke röstet un café in Владивосток!"
let lowerCase = NSCharacterSet.lowercaseLetterCharacterSet()
let upperCase = NSCharacterSet.uppercaseLetterCharacterSet()

for currentCharacter in testString.utf16 {
  if lowerCase.characterIsMember(currentCharacter) {
    println("Character code \(currentCharacter) is lowercase.")
  } else if upperCase.characterIsMember(currentCharacter) {
    println("Character code \(currentCharacter) is UPPERCASE.")
  } else {
    println("Character code \(currentCharacter) is neither upper- nor lowercase.")
  }
}

Swift 3

let testString = "Åke röstet un café in Владивосток!"
let lowerCase = CharacterSet.lowercaseLetters
let upperCase = CharacterSet.uppercaseLetters

for currentCharacter in testString.unicodeScalars {
    if lowerCase.contains(currentCharacter) {
        print("Character code \(currentCharacter) is lowercase.")
    } else if upperCase.contains(currentCharacter) {
        print("Character code \(currentCharacter) is UPPERCASE.")
    } else {
        print("Character code \(currentCharacter) is neither upper- nor lowercase.")
    }
}