Based on your question, your repository now looks like this:
o
^
feature
and you want to make it look like this:
o-------o
^ ^
master feature
while also making master the new default branch on GitHub, instead of feature.
One way to go about it is to create a new "initial" commit on the feature branch, move it to the beginning of the history and create a new master branch that points to it.
Here are the steps:
git checkout feature
git commit --allow-empty -m "Initial commit" (you can create the README file here, instead)
git rebase -i --root
# The TODO file will look something like this:
pick 1234abc Adds the feature
pick 5678edg Initial commit
# Move the "Initial commit" line to the top of the file
pick 5678edg Initial commit
pick 1234abc Adds the feature
# Then save and close
At this point, your history will look like this:
o-------o
^
feature
Now, create the master branch pointing to the commit before the one referenced by feature:
git checkout -b master feature^
At this point, all you have to do is push master to GitHub with:
git push -u origin master
Finally, you'll have to go to your repository's settings on GitHub to make master the new default branch. See GitHub's documentation on how to do that.