There are two ways of cloning a repository, either using HTTPS or using SSH.
The difference is that HTTPS is typically used to clone public third-party repositories, because it will not verify any credentials as long as the repository that you are attempting to clone is public.
Cloning it via SSH on the other hand will require you to first register your Private Key on GitHub, which it will use for Authentication. This is typically used for any repository that you want to modify - as well as any non-public repositories.
AFAIK, there used to be a way of also pushing changes via HTTPS - but it has always been highly discouraged to do so for security reasons. I have no idea whether that is even still supported.
The easiest thing you can do is to first set up SSH access to your repository and then push your changes via SSH.
There is no need to delete your repository and re-checkout, you can simply either add a new remote to it, or change the URL on the existing one.
For instance, once you've set up SSH, look at the SSH Clone URL and then you can do
git remote add ssh <your-ssh-remote>
git push -u ssh master
or
git remote set-url origin <your-ssh-remote>
git push -u origin master