Firstly, if you try to use a popup in your manifest.json and find out that your chrome broweraction onclick listener doesn't seems to work. You are in the right place then. And if you read properly on Google Chrome Extension API page, the following onclick will not work properly with popup if you have define it on your manifest.json file.
chrome.browserAction.onClicked.addListener(function(Tab tab) {...});
Fired when a browser action icon is clicked. This event will not fire if the browser action has a popup.
Well, i have another solution for you if you want both popup and onclick on Google Chrome extension to work together.
Popup and onclick working together on Google Chrome extension
Here is probably what you are doing. You are most likely placing your onclick event handler on your background.html file and awaiting for click of your chrome extension to do something within the onclick. And maybe sending them to your popup display. You soon realize that if the popup options is set on your manifest, your onclick won't work. Now, in order to make everything work, just use the popup instead of both as popup will overwrite the onclick event handler and you want the display to be out. What you should do is to paste the code from your background.html to your popup as you shouldn't have the need for a background.html as you are not communicating with many extension. Therefore, you should really either use popup or an onclick but not both 🙂
You might have guess, i was the dumb one here who did the above and realize i'm dumb. *shy* . hahaha. Hope it helps 🙂