This question is difficult to answer because its topic is so broad.
But to answer if it's possible. Yes it is possible. You'll see that there are already RSS reader applications out there, go and check the webstore and even "reverse engineer" them to study it up if you want.
I also did some RSS reader for a few clients before.
I hope I can give you sample codes to you now but its better if you start looking into main documentation and sample apps.
You should read on XHR to access pages
https://developer.chrome.com/extensions/xhr
You may encounter a lot issues in tackling this project.
- You need to consider that websites may load slow and your app should provide some UI for loading and some timeout to give up after a few seconds.
- Webpages will redirect, so you should handle if you are going to follow it or not
- Sometimes link is 404, or sometimes its not XML
- You need to cache things, you may need servers help for this
It's a tough project to do but don't get discouraged.
It is still a fun and challenging one and you should go for it still!
Have fun coding!