create new github repository with existing repository without fork

Clone existing repo into new project folder (e.g., my-proj).

git clone  https://github.com/ishara/my-proj  my-proj
cd my-proj

Delete everything "git-like" by deleting the .git folder.

rm -rf .git

Create a new git repo

You can writing code now. If you'd rather preserve your work under source control, consider taking the following steps.

Initialize this project as a local git repo and make the first commit:

git init
git add .
git commit -m "Initial commit"

Create a remote repository for your project (https://github.com/new).  

Get its address (e.g. https://github.com/<my-org>/my-proj.git) and push the local repo to the remote.

git remote add origin <repo-address>
git push -u origin master