Clone repo and set it up as new repo on your own GitHub

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:

$ git remote set-url origin http://github.com/YOU/YOUR_REPO

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

Or maybe even call the old one upstream:

$ git remote rename origin upstream
$ git remote add origin http://github.com/YOU/YOUR_REPO

Then, whenever you want to get changes from upstream, you can do:

$ git fetch upstream