Xcode 6.2 (and possibly 6.x) changed things around. There no longer is a Repositories tab in the Organizer. Here's how to do it with the newer Xcodes.
To add a project with a local git repository to GitHub, this is what you do:
- Go to the Source Control menu, elect your project and select Configure:

- You'll see:

- Select the Remotes tab, and press the "+" in the lower left corner. Select Add Remote.
- Type in a name for the remote repository, and type in the URL for your remote repository. Make sure you've created this remote repository already, and if on GitHub, make are you specify that this is an existing project, and not a new one.

- You're almost done. Go to the Source Control menu and select Push. You will then see the name of the repo you used in the last step, something like "simplest/master"; press the Push button.
Done.