Update: I still got befuddled by VLC but have successfully used Handbrake to do the conversions. Happy days. I have also got Netflix to cast from my phone to the TV - check out my 21st century skillz. It also explains the lack of washing up, hoovering, ironing.....
I got my Chromecast! And it all works and everything...to a point. It won't play .avi videos. Fine. I can use VLC to convert .avi to .mp4, which Chrome will handle.
So I did, and felt all smug, and then suddenly realised the converted film was completely mute. I can't find anything on Google that relates to the recent version of VLC and/or is understandable. There are things about codecs and finding errors in codes and...eeep.
I've fiddled with the VLC options in terms of trying the default conversion "Video - H.264 + MP3 (MP4)". I also tried a "Video for MPEG4 TV/device" which did precisely nothing.
I present you with thanks and bribery in the form of gifs...