How to add a privacy-checkbox in comment-template?

You can use the provided filters to do all that:

//add your checkbox after the comment field
add_filter( 'comment_form_field_comment', 'my_comment_form_field_comment' );
function my_comment_form_field_comment( $comment_field ) {
    return $comment_field.'<input type="checkbox" name="privacy" value="privacy-key" class="privacyBox" aria-req="true"><p class="pprivacy">Hiermit akzeptiere ich die <a target="blank" href="http://wp.cloudstarter.de/?page_id=156">Datenschutzbedingungen</a><p>';
}
//javascript validation
add_action('wp_footer','valdate_privacy_comment_javascript');
function valdate_privacy_comment_javascript(){
    if (is_single() && comments_open()){
        wp_enqueue_script('jquery');
        ?>
        <script type="text/javascript">
        jQuery(document).ready(function($){
            $("#submit").click(function(e){
                if (!$('.privacyBox').prop('checked')){
                    e.preventDefault();
                    alert('You must agree to our privacy term by checking the box ....');
                    return false;
                }
            })
        });
        </script>
        <?php
    }
}

//no js fallback validation
add_filter( 'preprocess_comment', 'verify_comment_privacy' );
function verify_comment_privacy( $commentdata ) {
    if ( ! isset( $_POST['privacy'] ) )
        wp_die( __( 'Error: You must agree to our privacy term by checking the box ....' ) );

    return $commentdata;
}

//save field as comment meta
add_action( 'comment_post', 'save_comment_privacy' );
function save_comment_privacy( $comment_id ) {
    add_comment_meta( $comment_id, 'privacy', $_POST[ 'privacy' ] );
}