How to implement chained method calls like jQuery?

You are almost there:

new foo('hello').alertTest('world');

or if you don't like the new:

var bar = function bar(str) {
    this.str = str;    
};

bar.prototype = {
    alertTest :  function(additional){
        alert(this.str + ' ' + additional);
        return this;
    }
};

function foo(str) {
    return new bar(str);
}

foo('hello').alertTest('world');

Live Demo.

I did something like this a while ago and it was a ton of fun to create!

If i remember correctly, To be able to use dot-operators, I had to return the object as part of the original function call. This way I could chain lots of stuff together like $(id).value('asdf').color('#ff0000')

function $(id){
    this.e = document.getelementbyid(id)
    me = this
    this.val = function (newval) {
        this.e.value = newval;
        return me;  // <- Important
    };
    return this;  //  <- Important
}

$("textbox1").val("New Value")    // changes textbox1's value to "New Value"

If it helps for reference: http://www.mikedoesweb.com/vis/