Travis-CI should automatically deploy the website a few minutes after the branches are merged or pushed on master, so you shouldn't need to worry about the building process.
If you would like to build/push manually though, see next section.
To manually build the website in the gh-pages directory, use
$ make gh-pages
This will clone the built-website repository, build a new version in it, auto-commit, and will then prompt you to check that everything is fine before pushing. Of course you need to have the right to push.
Additionally run
$ make linkcheck
if you have time to spare. This will launch a check of all external links.
Any fixes to the website, including this README, should be made in the ipython-website repository.