First, duplicate the repo (details here):

Create a new repo (let's call it private-repo) via the Github UI. Then:

git clone --bare https://github.com/exampleuser/public-repo.git
cd public-repo.git
git push --mirror https://github.com/yourname/private-repo.git
cd ..
rm -rf public-repo.git

Clone the private repo so you can work on it:

git clone https://github.com/yourname/private-repo.git
cd private-repo
make some changes
git commit
git push origin master

To pull new hotness from the public repo:

cd private-repo
git remote add public https://github.com/exampleuser/public-repo.git
git pull public master # Creates a merge commit
git push origin master

Awesome, your private repo now has the latest code from the public repo plus your changes.

Finally, to create a pull request private repo -> public repo:

Use the GitHub UI to create a fork of the public repo (the small "Fork" button at the top right of the public repo page). Then:

git clone https://github.com/yourname/the-fork.git
cd the-fork
git remote add private_repo_yourname https://github.com/yourname/private-repo.git
git checkout -b pull_request_yourname
git pull private_repo_yourname master
git push origin pull_request_yourname

Now you can create a pull request via the Github UI for public-repo, as described here.

Once project owners review your pull request, they can merge it.

Of course the whole process can be repeated (just leave out the steps where you add remotes).