First off, if you have a default_popup defined in the manifest - you need to remove it, as it interferes with the click event you want to catch.

Then, you need to catch the event in a background script:

chrome.browserAction.onClicked.addListener(function(tab) {
  // ...
});

Next, if we want a window, we probably want to look at the windows API. create() sounds like what you need:

chrome.browserAction.onClicked.addListener(function(tab) {
  chrome.windows.create({/* options */});
});

What options do you need? Assuming you want to open a page from your extension, you'll need an URL wrapped in a chrome.runtime.getURL:

chrome.browserAction.onClicked.addListener(function(tab) {
  chrome.windows.create({
    // Just use the full URL if you need to open an external page
    url: chrome.runtime.getURL("mypage.html")
  });
});

Then, to show a window like you're showing, without top toolbar, you need a window type "popup":

chrome.browserAction.onClicked.addListener(function(tab) {
  chrome.windows.create({
    url: chrome.runtime.getURL("mypage.html"),
    type: "popup"
  });
});

Finally, if you want to do something after the window has opened, use the callback:

chrome.browserAction.onClicked.addListener(function(tab) {
  chrome.windows.create({
    url: chrome.runtime.getURL("mypage.html"),
    type: "popup"
  }, function(win) {
    // win represents the Window object from windows API
    // Do something after opening
  });
});