Having a private branch of a public repo on GitHub?

I have a public PHP project in a GitHub repo, which contains just one branch (master).

I want to have a separate branch/fork that is private for me (I have paid for private GitHub repos). I would like to be able to merge changes from the private branch/fork to the public repo, and vice versa.

With that in mind, here are my questions:

1. Can I have a private branch on a public repo?
2. Can I fork my own public repo into my own private branch/fork?
3. If both of the above are possible, which is the best way forward? If neither, how should I proceed?

1. Duplicate your public repo.
2. Make the duplicated repo a private one on GitHub.
3. Clone the private repo to your machine.
4. In this local clone of your private repo:
   1. Add a remote to your public repo: git remote add public git@github.com:...
   2. Push branches with commits intended for your public repo to that new public remote. Make sure you don't accidentally commit private-only code.
   3. You can bring in changes from your public repo using git fetch public and then merge them locally and push to your remote private repo (origin remote).