Store a reference to the original function in a variable:
(function() {
    var _alert = window.alert;                   // <-- Reference
    window.alert = function(str) {
        // do something additional
        if(console) console.log(str);
        //return _alert.apply(this, arguments);  // <-- The universal method
        _alert(str);                             // Suits for this case
    };
})();
The universal way is <original_func_reference>.apply(this, arguments) - To preserve context and pass all arguments. Usually, the return value of the original method should also be returned.
However, it's known that alert is a void function, takes only one argument, and does not use the this object. So, _alert(str) is sufficient in this case.
Note: IE <= 8 throws an error if you try to overwrite alert, so make sure that you're using window.alert = ... instead of alert = ....