How best to install/upgrade software outside the official repo

Download the .deb file
and install as demonstrated (for some other random package) here

Needless to say that there are security and trust issues in installing random packages from the internet.

If it fails to install, please post error messages.

The most likely errors are missing dependencies. There is no easy way to sort that out but a starting point would be to check with the web site where you downloaded the .deb file what the expected version of Debian (or other Linux distro) is.