How can I backup my iMac with a bad video card?

It's eligible for a free replacement according to this program:

https://support.apple.com/en-us/HT203787

I need to turn it on and back it up. I managed to start it up yesterday and it backed up 120/460 GB. Now I can't start it up by following the same steps.

I need to backup the data before I go and replace it.

It's urgent. I have time sensitive files.