Javascript on click event not reading else statement or variables

Every time you click you are actually reseting the string. Check the scope!

var str = "Learning to Code Javascript Rocks!";
var button = document.getElementById("button");
var output = document.getElementById("output");


output.innerHTML = str;

button.addEventListener("click", function(e){
	str = str.charAt(str.length - 1) + str.substring(0, str.length - 1);
  output.innerHTML = str;
});

button{
    display: block;
    margin: 25px 0;
}

<button id="button">Click Me!</button>

<label id="output"></label>

It is, in fact, a scoping issue. Your counter in inside the function, so each time the function is called, it gets set to 0. If you want a counter that is outside of the scope, and actually keeps a proper count, you will need to abstract it from the function.

If you want to keep it simple, even just moving clickCount above the function should work.

I do not know if it's an issue of scope

Yes, it is an issue of scope, more than one actually.

How?

1. As pointed out by @thesublimeobject, the counter is inside the function and hence gets reinitialized every time a click event occurs.
2. Even if you put the counter outside the function, you will still face another scope issue. In the else part of the function, you are manipulation a variable (newString) you initialized inside the if snippet. Since, the if snippet didn't run this time, it will throw the error undefined. (again a scope issue)

A fine approach would be:

1. take the counter and the defaultString outside the function. If the defaultString gets a value dynamically rather than what you showed in your code, extract its value on page load or any other event like change, etc. rather than passing it inside the function.
2. Do not assign a new string the result of your manipulation. Instead, assign it to defaultString. This way you probably won't need an if-else loop and a newLastChar to take care of newer results.
3. Manipulate the assignment to the element accordingly.

You can use Javascript closure functionality.

var scrollString = (function() {
var defaultString = "Learning to Code Javascript Rocks!";
return function() {
    // convert the string into array, so that you can use the splice method
    defaultString = defaultString.split('');

    // get last element
    var lastElm = defaultString.splice(defaultString.length - 1, defaultString.length)[0];

    // insert last element at start
    defaultString.splice(0, 0, lastElm);

    // again join the string to make it string
    defaultString = defaultString.join('');
    document.getElementById('Result').innerHTML = defaultString;
    return defaultString;
}
})();

Using this you don't need to declare any variable globally, or any counter element.

To understand Javascript Closures, please refer this: http://www.w3schools.com/js/js_function_closures.asp