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.
Though, if you like to build/push manually see next section.
Use make to build the website in the gh-pages directory, use
$ make gh-pages
It will clone the built-website repository, built a new version in it, and auto-commit. 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 loose fixing old link.