How to upload a cloned git repository to an own git repository on github?

I have cloned a project from github. Now I want to upload it to my own private repository on github to work on it. But the cloned repository has in the origin key the value of the cloned project. What's the best practice to upload this project to my own private repository? I've read the german version of the git documentation (https://git-scm.com/book/en/v2/Git-Basics-Working-with-Remotes) but haven't found the solution.