I would suggest you to use Selenium. Since it is an open-source tool, you’ll even find some assistance to work on the process.
Meanwhile, there is this guide by Browserstack that you might find helpful on your way to test extensions.
Chrome extensions are HTML/CSS/JS based applications. You can test them with Selenium, the steps look something like this:
Build the extension to yourdir
Start browser and load extension with the argument load-extension=
Open url chrome://extensions/
Find the ID of extension under test
Open url chrome-extension://ID/options.html to test/configure the option ...