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

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

Or maybe even call the old one upstream:

$ git remote rename origin upstream
$ git remote add origin

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

$ git fetch upstream