How do I include background images in my stylesheets in a plugin?

Just add a STYLE tag in 'wp_head'.

// Anonymous function, PHP 5.3+, extract function for 5.2
add_action('wp_head', function(){
    echo '<style type="text/css">';
    // Add background style here
    echo '</style>';
});

Add this in your functions.php. Don’t stuff too much CSS in it, just the dynamic bg image you need.

It’s not the easiest thing to create a dynamic CSS file so, just for a bg image, use this method.