How to use jquery in google chrome extension page action background.js?

The "background" specification in manifest.json should specify jquery.js so that it is loaded before background.js:

...
"background": { "scripts": ["jquery.js","background.js"] },
...

This should do the job.
Remember the js files are loaded in the order they are specified.

test if jquery is loaded.

in background.js

if (jQuery) {  
    // jQuery loaded
} else {
    // jQuery not loaded
}