One way, you an treat this as a single one time request to the content-script which will fetch the dom you want to access.
http://code.google.com/chrome/extensions/messaging.html#simple
Basically, your content script sets up the listener:
chrome.extension.onRequest.addListener(
function(request, sender, sendResponse) {
if (request.greeting == "hello")
sendResponse({farewell: "goodbye"});
else
sendResponse({}); // snub them.
});
And your background page sends a single lived request:
chrome.tabs.getSelected(null, function(tab) {
chrome.tabs.sendRequest(tab.id, {greeting: "hello"}, function(response) {
console.log(response.farewell);
});
});
When you send your response, you send it as JSON data, you can fetch whatever you want (such as html, dom, text, etc).
That is currently the only way to let the background page know anything about the contents of a page. Remember you would need content scripts and tab permissions.