Generally speaking you'll want to create your own custom plugin, so start with a tutorial there to get a feel for the process. You basically just need to create a PHP file within a folder within
/plugins/ with a few comments to help WP recognize it as a plugin. From there you can start working on JS and enqueue it using your plugin only on the page you're searching from. Add a listener on the search input box, then either immediately look up results when input is added, or wait until they type say 3 characters and then look up results. Once results are found, append a div with CSS to overlay it underneath the search input.