Q:

Creating a new git repository from an existing one

I cloned a GitHub repository from an existing project.  Now I need to create a new separate BitBucket repository to push my code modifications to. 
How would I do this?  Create a sub-module of my current local git repository or just push the existing master to the new repository? 

A:

You can have any number of remotes you wish with a git repository.  This means you can just add the bitbucket remote and push your code as usual and it will work fine. 
You can define to push to which remote by using the following push syntax:
git push <remote> <branch>

And the same with fetch and pull:
git fetch <remote>
git pull <remote>

For further information you can look into the remote chapter of Pro Git.  And of course the documentation of push, fetch and pull.