- You weren't populating the rotated array. You were modifying the array reference!
- You were checking if index == 4 when you could do array.length - 1.
- You need to determine the target index as being (index + rotationTimes) then if it exceeds array.length - 1 (the last index) you use the modulus operator to make sure it doesnt overflow. For example, if the current index is 4 (the last one) and the iterations is 1, (4 + 1) is greater than 4 so the target index is (4 + 1) % 5 = 0. 
    public static void main(String[] args) {
        int[] array = new int[] {1, 2, 3, 4, 5};
        int[] rotated = new int [array.length];
        System.out.println("Enter amount of times to rotate: ");
        Scanner scan = new Scanner(System.in);
        int rotationTimes = scan.nextInt();
        if (rotationTimes > 4){
            rotationTimes %= 5;
        }
        for (int index = 0; index < array.length; index++) {
            int targetIndex = index + rotationTimes;
            if (targetIndex > array.length - 1) {
                targetIndex %= array.length; 
            }
            rotated[targetIndex] = array[index];
        }
        System.out.println("Input:  " + Arrays.toString(array));
        System.out.println("Rotations: " + rotationTimes);
        System.out.println("Output: " + Arrays.toString(rotated));
    }
Input:  [1, 2, 3, 4, 5]
Rotations: 1
Output: [5, 1, 2, 3, 4]
Input:  [1, 2, 3, 4, 5]
Rotations: 2
Output: [4, 5, 1, 2, 3]