How to backup with Time Machine in Single User Mode?

Problem resolved:

(You will need another mac with a Thunderbolt port to make this work. We'll call the problematic one A and the other one B)

1. Connect a Thunderbolt cable between the old machine (A) to another one (B).
2. Start A in target mode (Press T on boot).
3. You'll now see A's drive as an external drive on B's desktop. Go to 'Startup Disk' in 'System Preferences'. Pick A's drive as your boot disk.
4. Reboot. If everything's working, you should see A's Finder etc. on B's display.
5. Connect Time Machine to B. Start it. Click 'Options...' and exclude B's drive from the backup.
6. Run another time machine backup on the HD.

I eventually went out and got a new MBP (which I was planning to do anyway) and migrated directly from the time machine latest backup.

Using the command line you can do the rsync as it only copies files changed since the last time you ran it.

rsync -av "/Users/your name" "/Volumes/My External Drive"