Chromecast does not have any inbuilt option to configure VPN on the device itself. There is a workaround to overcome this inability of Chromecast to be configured with any VPN, you can connect it to a router which is already configured with a VPN.
Below are the steps you need to follow to run Chromecast over VPN after configuring your router with LimeVPN:
Navigate to the “Connect Chromecast to your Wi-Fi”page.
From the list of available Wi-Fi networks. select the router-network which is configured with LimeVPN.
Enter the password for which you have set for your router, then hit on "SET NETWORK" button.
If everything went well you will get connected and now you are ready to cast.