Read this SO question in order to understand the difference between content script and background script. Now, back to your question
Since you want to get access to the DOM element of the current page, you need to do the following:-
In your manifest.json, make sure you have the url pattern which matches the current page url.
Make your your content script runs after page load on the current page
Once you ensure 1 and 2, you can use simple query selectors to get access to the table element and get the values you need.
Now it depends what you do want to do with the table values you have obtained. If you need them in the background script for storage purpose or some other purpose, you can use Mesage Passing to send the values to background script.