How do I store data from an ajax call?

Since you’re using jQuery, you can easily access the HTML DOM.

Consider an iframe element in your HTML:

<iframe src="" width="1920" height="1110" style="border: none;"></iframe>

You can interact easily with jQuery:

$('iframe').attr('src', 'http://www.twitch.tv/' + stream.channel.name + '/embed');

Information on jQuery selectors: https://api.jquery.com/category/selectors/