getCurrentTime() for YouTube Video

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// WARNING: This example is out of date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

Use the following code works for chrome extension:

video = document.getElementsByClassName('video-stream')[0];
console.log(video);
console.log(video.currentTime);

In 2020, it seems we should use: player.playerInfo.currentTime.

full code on codepen