First rename the old remote as upstream, in case you want to be able to keep in sync with the original repository.
git remote rename origin upstream
Then add your forked repository as origin:
git remote add origin https://github.com/<your-username>/<your-project>
Or if you’re using ssh:
git remote add origin git@github.com:<your-username>/<your-project>.git
To push to your repository:
git push -u origin master
To pull from the base repository:
git pull upstream
I recommend you do all of your work in a separate branch, not the master branch. It will be easier to rebase to the upstream/master branch in case you want to make a pull request.
You don’t really have to rename the origin to upstream – the remote names can be arbitrary, however I recommended you to do so to keep up with the naming convention used by GitHub.