How to upload existing local project to a remote existing and empty github repository
When you create a new repository in Github, all the steps required are shown:

You just need to modify this accordingly for your use case:

you already have files, so no creating files
you may want to create the correct .gitignore for your project type, usually you exclude temporary files, such as caches, logs as well as files containing credentials
For this to work as per SSH example, you already need your primary public key uploaded to the same github account. If not, you can do this with a Github token, or look at the https instructions
if the existing repository already has commits in it, are you supposed to wipe them out? In this case use -f on git push.