Adding a full screen background image to Drupal 7 / Bartik theme
Are you actually trying to use fullscreenr because you want a full screen background image that re-sizes with the browser? Or do you just want to have a background image?

If you're just looking to use a background image, you don't need jQuery. You can just edit the CSS. Though, if you're changing a builtin theme like Bartik, you should actually create a subtheme instead of editing the builtin theme's files.

If you actually need fullscreenr then you need to add the javascript files for the plugin and your file that calls the fullscreenr method to your theme's .info file (again, you should not edit the builtin theme's files, you should create your own subtheme). In the example below, I'm just editing Bartik's files (don't do this, use a subtheme). You should add the following to the theme's .info file

stylesheets[all][] = jquery/fullscreenr/css/fsbg-style.css
scripts[] = jquery/fullscreenr/js/jquery.fullscreenr.js
scripts[] = js/fullscreenbg.js


In this example, I added a jquery folder to the theme's folder and put the fullscreenr folder containing the plug-in into the jquery folder. I also slightly renamed the style.css file that comes with the plugin (in addition to deleting all the extra option stuff from that file). I put my code for actually calling the plug-in into the js/fullscreenbg.js file. The code looks like (details on why it's wrapped like that in the Drupal docs for changes from D6 to D7):

(function ($) {
  // You need to specify the size of your background image here (could be done automatically by some PHP code)
  var FullscreenrOptions = { width: 1024, height: 768, bgID: '#bgimg' };
  // This will activate the full screen background!
  jQuery.fn.fullscreenr(FullscreenrOptions); 
}(jQuery));    


I think this was the part you were really having trouble with.

There are some other minor things you need to do to set up the fullscreenr plug-in. I highly suggest getting it to work on a very basic html page outside of Drupal before putting it into a Drupal theme so you know you're at least using the plug-in correctly. Also, keep in mind that Bartik has a lot of other stuff going on with background colors and such on various parts of the page, so just adding the background image without editing the theme's CSS is not going to appear to make any difference.