Fork the repo on GitHub, then add your fork repo as a remote to your local cloned copy:
git remote add myfork https://github.com/<myGitHubAccountName>/<repoName>.git
Then you can push to your fork:
git push myfork master
If you’re doing more than just this one pull request, you can remove the origin remote and name your fork as origin:
git remote rm origin
git remote add origin https://github.com/<myGitHubAccountName>/<repoName>.git
This is typically what I do. Sometimes I add the original origin as upstream so I still have a reference to it.