How to add a class to ul tags created by the WordPress editor Tinymce?

It’s always a good idea to look at WordPress Codex before asking. TinyMCE Custom Styles

<?php
// Insert 'styleselect' into the $buttons array
function my_mce_buttons_2( $buttons ) {
    array_unshift( $buttons, 'styleselect' );
    return $buttons;
}
// Use 'mce_buttons' for button row #1, mce_buttons_3' for button row #3
add_filter('mce_buttons_2', 'my_mce_buttons_2');

function my_mce_before_init_insert_formats( $init_array ) {
    $style_formats = array(
        array(
            'title' => 'Custom UL class', // Title to show in dropdown
            'selector' => 'ul', // Element to add class to
            'classes' => 'custom-ul-class' // CSS class to add
        )
    );
    $init_array['style_formats'] = json_encode( $style_formats );
    return $init_array;
}
add_filter( 'tiny_mce_before_init', 'my_mce_before_init_insert_formats' );

In editor: first create an unnumbered list, then apply the style.

Update after acceptance.

If you want to append a class to all <ul> elements added through TinyMCE you can do it prior to inserting into or updating the database:

<?php
add_filter('wp_insert_post_data', 'my_add_ul_class_on_insert');
function my_add_ul_class_on_insert( $postarr ) {
    $postarr['post_content'] = str_replace('<ul>', '<ul class="my-custom-class">', $postarr['post_content'] );
    return $postarr;
}

Tune str_replace() function needle if <ul> elements already has any attributes. Or use preg_replace() there.