You can push from the original repository to the cloned repository. To do so you must first configure a "remote" in the original repository:
git remote add backup //backupnas/backup/GIT/TexRepo.git
Now you can push from the original to backup:
git push backup
To keep the clone updated you will have to push regularly.
I do not recommend this but it is possible to automate a push after every commit. Here is an example how to do that.
I consider a push after every commit as going against the designed workflow of git, which specifically allows to rewrite your local history (which hasn't been pushed yet). Read this to learn more about this feature of git.
Think of push as one big commit. You work all day doing local commits and the occasional rewriting of local history. Then, at the end of the day you do one big commit, a push.