You can open the background page’s console if you click on the “background.html” link in the extensions list.
To access the background page that corresponds to your extensions open Settings / Extensions
or open a new tab and enter chrome://extensions
. You will see something like this screenshot.
Under your extension click on the link background page
. This opens a new window.
For the context menu sample the window has the title: _generated_background_page.html
.