1.) Is it possible to have a private branch on a public repo
From what I know, no.
2.) Can I fork my own public repo into my own private branch
No, you can't fork a full repo (1-n branches) into a single branch. Well actually you could, if you just fork the one branch of the full repo. Just add it as a remote or start from a clone.
Is it possible to have a private branch on a public repo?
On GitHub, your repository is either public or private; you cannot selectively "privatize" just a branch.
Can I fork my own public repo into my own private branch/fork?
You can clone your public repo to your local machine, branch as needed, and simply not push your "private" branches upstream (by specifying which branch to push to origin: git push origin master or git push origin branch-i-want-to-be-public:master).
Which is the best way forward/how should I proceed?
In order to take advantage of GitHub for both your public and private development, I would suggest forking your public branch within GitHub, changing the settings of the new fork to "Private", and then cloning the private version down to your local machine. When you're ready to make changes public, push everything up to your private fork on GitHub and then use pull requests to selectively copy branches to the public repo.
To make a repository private on GitHub, you must have an upgraded (paid) account. If you're only rocking the free account, you can still use the first process I suggested — clone public to local machine, branch, and push specific "public" branches to origin — without needing a private repo.
If you have a paid GitHub account, or are using another service that offers public and private forks and pull requests (such as BitBucket), then you can use either of the above approaches to make your code public.