Travis-CI should automatically deploy the website 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.