I think that the "most polite way" to do so would be:
Fork the original repo on your GitHub account
Checkout a new branch for your changes git checkout -b <your_branch_name> (in case you didn't do that before)
Add a new remote for your local repository: git remote add github <your_repository_ssh_url>
Push your beautiful new branch to your github repository: git push github <your_branch_name>
In this way you will have a repo forked to the original one, with your changes commited in a separate branch. This way will be easier in case you want to submit a pull request to the original repo.
As Deefour says, your situation isn't much unlike the one in Change the URI (URL) for a remote Git repository. When you clone a repository, it is added as a remote of yours, under the name origin. What you need to do now (as you're not using the old source anymore) is change origin's URL:
If the original repository would update often and you want to get those updates from time to time, then instead of editing origin it would be best to add a new remote:
$ git remote add personal http://github.com/YOU/YOUR_REPO
Then, whenever you want to get changes from upstream, you can do:
$ git fetch upstream
As this the source is a sample repository (seems to be kind of a template to start off), I don't think there's a need to keep it nor fork it at all - I'll go with the first alternative here.
Carry on as normal there and you can't be tripped up by issues with git. I do this fairly often. Certainly not as 'pure' as the other answers, but trauma free and removes any potential problems you may have.
Using vscode, these other solutions did not quite work for me,
Instead, I ran
git init
This will make it so that the 'new branch' button makes a completely new repository when you go to commit. I could not figure out how to commit into an already uninitialize repository.