You can only install apps in the new Chromecast with Google TV device (the oblong one that comes with a remote).
You can't do that with any of the other Chromecast devices.
The easiest way to install apps to Chromecast with Google TV is to use the web browser version of the Play Store. On your computer, go to https://play.google.com and login to your Google Account. That can push apps to your Chromecast. NOTE: not all apps work with Chromecast so you will see a lot that say "this app is not compatible with your device."
Also, the Chromecast with Google TV does not have a lot of memory in it. It can't hold a lot of apps.
From: Byteguy
(Gold Product Expert)