I know the android app called "filmon tv", it is great and with it I can watch UK and USA TV channels on my smartphone. I have also a chromecast device on my TV and wonder how could I watch these "filmon tv" channels on my tv since I do not see any option on the app.
A Chromecast version of FilmOn is now available in the Play Store - it works perfectly!
If you have a Nexus 4, 5, 7, or 10, or some "Google Play edition" phones, running KitKat, then you can simply cast the whole screen of your device. In the settings app, go to Display, then click Cast screen. This can cast to any Miracast device. This might not work with the particular app you want to use: the app can tell Android that its content is "secure", which will exclude it from being recorded or cast in this way.
One of the CWM developers has an app called Mirror. This can cast the whole screen to a Chromecast device, but it's in beta, so you have to register to download it, and it only works on the Nexus 5. See his announcement on G+ for more information and to register.
Otherwise, all you can do is contact the developer and ask them to add Chromecast support to their app. This would be the best option in any case, as the app could make sure that only the movie stream is sent to the Chromecast, rather than the whole screen (including the notification tray and navigation buttons).