How can I send the video to an external screen?
KaraFun fully supports Chromecast. You can use it to transfer both audio and video contents on your TV or any other display that has a Chromecast plugged in (you will need a Chromecast stick).
However, bear in mind that Chromecast technology is still young and not all devices support screen sharing through Chromecast (see list of compatible devices here). You need a compatible device and Android 4.4.2 and above.
Follow the steps described here to learn how to cast your screen.
When you're done, KaraFun should show up on your TV and you can use KaraFun Android as usual, with the lyrics appearing on your TV.