ChatZilla is normally used as part of SeaMonkey, or an extension to other browsers. However, it can also run under XULRunner, allowing ChatZilla to be used standalone, independent of a web browser.
Run the installer and start ChatZilla from the Start Menu/Screen, or extract the .zip package and run chatzilla.exe.
Simply extract a tarball and run chatzilla.
These steps have been reported broken.
If you need to copy over settings or scripts, use the ChatZilla command /pref profilePath to find your profile.
ChatZilla will regularly check for updates, but nothing will be downloaded without your permission. If you downloaded a package including XULRunner, then that will also be included in the updates.
In addition to ChatZilla plugins, these builds support extensions and themes. The Extension manager can be opened from the ChatZilla menu.
You don't need to build Mozilla to build ChatZilla. You just need Mercurial, Python, and something to apply a diff.