Is there a way to entirely remove a directory and its history from GitHub?
If you are asking about deleting a project from GitHub, open your project, click the "Admin" tab (or browse directly to https://github.com/username/project_name/edit) and on the bottom of the page, click "Delete This Repository". It'll ask you to confirm this, and then it's gone.
If you just want to erase a part of your repository, you need to do it to your git repository and push it to GitHub.
GitHub has written a howto about this in their FAQ. I Haven't tried this myself, so I can't guide you further, but you probably can manage this yourself here on.
In either case, this, naturally, doesn't delete any third party pulls – if someone has pulled the repository before you deleted it, it's out, without you being able to do much about it (other than trying the "pretty please"-technique).