Short answer: No, you cannot download or use Kik on Xbox 360. Kik Messenger is designed exclusively for mobile operating systems—iOS and Android. The Xbox 360 runs a completely different proprietary operating system with no compatibility layer for mobile apps.
If you want to chat with friends while gaming on Xbox 360, use the console's built-in messaging system accessible through the Guide button. You can send text messages, voice messages, and join party chat with Xbox Live Gold. For mobile-style chatting, keep Kik on your smartphone—it's where the app was designed to live.