You could use http://phpjs.org/ http://locutus.io/php/ it ports a bunch of PHP functionality to javascript, but if it's just echos, and the script is in a php file, you could do something like this:
alert("");
don't worry about the shifty-looking use of double-quotes, PHP will render that before the browser sees it.
as for using ajax, the easiest way is to use a library, like jQuery. With that you can do:
$.ajax({
url: 'test.php',
success: function(data) {
$('.result').html(data);
}
});
and test.php would be:
it would write the contents of test.php to whatever element has the result class.