Where does the JavaScript go?

The JavaScript code on a web page should go in a <script> tag. As for when to run it, for the JavaScript that creates the button, you'll probably want to run that when the page loads. How exactly you do that depends on what libraries you're using, if any, (e.g,. jQuery, etc.), so I can't answer that specifically for you.

Of, if you are using a custom button instead, the JavaScript to trigger the Chooser to appear should be run from the event caused by the user (e.g., when they click a custom button or like on your page).