Solución:
Código PHP:
<?php
define('APP_PATH', '/app');
require_once __DIR__ . '/vendor/autoload.php';
$request = KleinRequest::createFromGlobals();
$request->server()->set('REQUEST_URI', substr($_SERVER['REQUEST_URI'], strlen(APP_PATH)));
$klein = new KleinKlein();
$klein->respond('GET', '/hello', function () {
include 'hello.php';
});
$klein->dispatch($request);