How do I reformat and output a javascript string into array?

Question 1

Before anything else, if you are debugging, you are better off using the debugger of the browser (F12 in most browsers). Use console.log() to output to the console to see values. Breakpoints would be better since they pause code, and you can inspect values at that moment.

Now, for the mistakes you made:

Your output is test which is an <a>. You can't do a push since that's not an array. What you can do is create an array, fill it with values, and do a JSON.stringify to turn it into a string.
You should use document.get* functions, like document.getElementId() to refer to DOM elements. Though browsers do expose globals for elements with ids, you should avoid that.

This should fix it:

function alterText() {
    var myData = {
        "timestamps": [1369008000, 1369094400, 1369180800],
            "clicks": [1, 2, 3]
    };

    var output = [];
    var link = document.getElementById('test');

    for (var i = 0, l = myData.timestamps.length; i < l; i++) {
        output.push([myData.timestamps[i], myData.clicks[i]]);
    }

    // Output to console
    console.log(output);

    // Convert array to text and output to element
    link.innerHTML = JSON.stringify(output);
}

Question 2

A big problem with the jFiddle is you don't include jQuery. When trying to get an element by id, you cannot just reference it by name. Below is a revised version that does output what you want (though not in a nice format or anything).

$(function() {
    var myData = {"timestamps":[1369008000,1369094400,1369180800,],"clicks":[1,2,3,]};

    var output = $('#test'); 
    for (var i = 0; i < myData.timestamps.length; i++) { 
        output.append([myData.timestamps[i], myData.clicks[i]]);
    }
});