How to customize wp_link_pages()?

You can use next_or_number param of wp_link_pages to achieve this.

From Codex:

next_or_number

(string) Indicates whether page numbers should be
used. Valid values are:

• number (Default)
• next (Valid in WordPress 1.5 or after)

Then you can use nextpagelink and previouspagelink params to modify these links texts.

So doing it like this should do the job:

<?php
    wp_link_pages( array(
        'next_or_number' => 'next',
        'nextpagelink' => 'next &gt;&gt;',
        'previouspagelink' => '&lt;&lt; previous'
    ) );
?>