How to write to wp-users table

You need to use wp_update_user function to update user info. Please see below code:

<?php

$user_id = 1; /* It would be dynamic in your case */

$display_name = $_POST['display_name'];

$user_id = wp_update_user( array( 'ID' => $user_id, 'display_name' => $display_name ) );

if ( is_wp_error( $user_id ) ) {
    // There was an error, probably that user doesn't exist.
} else {
    // Success!
}

?>