You can't delete files over HTTP (well in theory you can, but it's not implemented.)
The easiest way is to set up a tiny server side script (e.g. in ASP or PHP) and to call that from JavaScript. The server side script needs the proper permissions to do the deletion, but otherwise there is no problem.
In PHP the start would look like this: (Not expanding solution to a fully secure one because you're not saying what platform you are on)
<?
// STILL INSECURE!!!!
// Do not use in any public place without authentication.
// Allows deletion of any file within /my/files
// Usage: filename.php?file=filename
$basedir = "/my/files";
$file_to_delete = $_REQUEST["file"];
$path = realpath($basedir."/".$file_to_delete);
if (substr($path, 0, strlen($basedir)) != $basedir)
die ("Access denied");
unlink($path);
?>
you would call the script like this:
http://yourserver/directory/delete_file.php?file=directory/filename