Twitter recently added support for custom Emoji on their web app. Their mobile apps already support them on most platforms. They have now added support for the same in their TweetDeck web app.
TweetDeck’s Chrome app is not yet updated with this new functionality but it is coming soon.
Google Chrome on Windows does not support emoji natively right now which is why users end up seeing boxes in place of the cute icons representing the emoji characters so this is a good new development.