As Ben correctly assumed, you're executing the code in the context of your background page, which is wrong.
To interact with other pages, you need to inject a content script in them. See overview of that here. You'll probably need to learn how Messaging works so you can gather data in a content script and communicate it to the background page.
To make a minimal example, you can have a file inject.js
// WARNING: This example is out of date
// For the current HTML5 player, see other answers' code snippets
ytplayer = document.getElementById("movie_player");
ytplayer.getCurrentTime();
And in your background page script, you can inject that into the current page as follows (when the button is clicked, for instance)
chrome.browserAction.onClicked.addListener(function(tab) {
chrome.tabs.executeScript({
file: 'inject.js'
});
});
Then, you'll see the result of the execution in the console of the currently open page. To report the result back, you'll need to use chrome.runtime.sendMessage