First, create a 'bare' clone of your existing repository to use as your backup:
$ git clone --bare /path/to/your/repo /path/to/your/backup_repo
Next, navigate to your existing repository add a new remote repository definition:
$ git remote add backup /path/to/your/backup_repo
The name of the remote definition in this instance is called 'backup' but you can call it whatever you like.
Now, whenever you're working on your existing repository you can push the changes to your new backup repository by simply executing the following command:
$ git push backup
You should see results something like this:
Counting objects: 5, done.
Compressing objects: 100% (2/2), done.
Writing objects: 100% (3/3), 287 bytes, done.
Total 3 (delta 0), reused 0 (delta 0)
Unpacking objects: 100% (3/3), done.
To /path/to/your/backup_repo
7c6f003..9eea051 master -> master
If you want to automate this process so you don't have to manually execute the push, you could probably just wire up a post-commit hook to do it for you.