Backup to Time Machine - not the root drive

Hi,

Wonder if anyone can help with the backup problem I am having.

My mac harddrive recently decided to fail on me not allowing me to boot the OS. After many, many hours I have managed to get the a fresh install of Mavericks to an external hard drive and boot from there. I can now see my original hard drive and all of the files that are on it which are all readable.

I now need to back this disk up to network storage so I can reformat the internal drive and restore the original contents. However, I have attempted to set up TimeMachine to backup the internal drive but it only wants to back up the external - where the OS has booted from. Please could someone point me in the right direction on how to back the data up safely? I am happy with any solution to back the data up as long I can easily restore once the internal disk has been formatted.

Thanks in advance
Dean