The way it's normally done is you fork the repo (press the "fork" button in the top right) then you have a copy of it on GitHub. Then you clone your repo to your local computer, and do all your work in there. Then you can check it in and push to your own repo as much as you like.
When you're happy with it you create a pull request (green button) which then generates a patch which I can apply to my repo automatically.