The simplest solution remain a branch within the repo, as mentioned by musicmatze.
Fork doesn't exist for one's own repo (and you can't have two repos with the same name): you can clone, and add an upstream remote as you would with a fork (see this gist), but you won't benefit from the pull request mechanism.
The only other solution would be to create a second GitHub account, and use it to (truly) fork repos from the first account.