[javascript] Chrome Extension - Get DOM content

The terms "background page", "popup", "content script" are still confusing you; I strongly suggest a more in-depth look at the Google Chrome Extensions Documentation.

Regarding your question if content scripts or background pages are the way to go:

Content scripts: Definitely
Content scripts are the only component of an extension that has access to the web-page's DOM.

Background page / Popup: Maybe (probably max. 1 of the two)
You may need to have the content script pass the DOM content to either a background page or the popup for further processing.

Let me repeat that I strongly recommend a more careful study of the available documentation!
That said, here is a sample extension that retrieves the DOM content on StackOverflow pages and sends it to the background page, which in turn prints it in the console:

background.js:

// Regex-pattern to check URLs against. 
// It matches URLs like: http[s]://[...]stackoverflow.com[...]
var urlRegex = /^https?:\/\/(?:[^./?#]+\.)?stackoverflow\.com/;

// A function to use as callback
function doStuffWithDom(domContent) {
    console.log('I received the following DOM content:\n' + domContent);
}

// When the browser-action button is clicked...
chrome.browserAction.onClicked.addListener(function (tab) {
    // ...check the URL of the active tab against our pattern and...
    if (urlRegex.test(tab.url)) {
        // ...if it matches, send a message specifying a callback too
        chrome.tabs.sendMessage(tab.id, {text: 'report_back'}, doStuffWithDom);
    }
});

content.js:

// Listen for messages
chrome.runtime.onMessage.addListener(function (msg, sender, sendResponse) {
    // If the received message has the expected format...
    if (msg.text === 'report_back') {
        // Call the specified callback, passing
        // the web-page's DOM content as argument
        sendResponse(document.all[0].outerHTML);
    }
});

manifest.json:

{
  "manifest_version": 2,
  "name": "Test Extension",
  "version": "0.0",
  ...

  "background": {
    "persistent": false,
    "scripts": ["background.js"]
  },
  "content_scripts": [{
    "matches": ["*://*.stackoverflow.com/*"],
    "js": ["content.js"]
  }],
  "browser_action": {
    "default_title": "Test Extension"
  },

  "permissions": ["activeTab"]
}