How to submit a pull request from a cloned repo?

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.

If you're ok with installing another binary in your path, github has released a nice little tool called hub.

If you've cloned someone else's repo:

$ hub fork  # This creates a fork and adds your repo as a remote

$ git push YOUR_USER feature  # push the changes to your new remote

$ hub pull-request  # will open your browser