1) You should not modify the screen.css file, because that would be a
hack to Blueprint's core files.
2) You should try to use a tool like Firebug, or WebKit's built-in Web
Developer Tools to debug this kind issues and ensure your resources
get delivered to the user's bowser as desired.
3) The screen.css file is one more directory deeper inside the atomic
template's css folder, so obviously, url(../images/template.jpg) won't
work. In such case the url(../../images/template.jpg) should be the
right one to use, but....
4) As I mentioned before in 1, this should go into the template.css
file under the css folder, so you may use Joomla's template editor to
change this file easily with the CodeMirror editor and keep the
background rule as background-image :url(../images/template.jpg);
5) Try reading w3schools.com
tutorials on CSS, specially when working
with relative URLs, and take a look to the Joomla's wiki articles on
templates design at docs.joomla.org
, then you are welcome to come back
later with more difficult questions ;-)