Once you have cloned this repository, why don't you push it back up to your own GitHub repo?
While logged in to GitHub, use the "+" icon in the upper right to create a new repository. Give it a name but leave it empty, don't even check the box to add a README. Then, change the remote url of this local clone to point to your GitHub origin. From a shell with working directory set to your local clone, enter:
git remote set-url origin https://github.com/%user%/%repo%.git
Replace %user% with your user name on GitHub and %repo% with the name of your new GitHub repo. Now just push the commits on all branches up to your origin:
git push origin --all
You are now able to commit and push any changes you make locally to your own repo on GitHub.