What to do about a bad video card in a 2010 Mac?
This answer is based on the assumption that you've got an iMac, however, the general solutions (not anything iMac specific) presented here will work for virtually any Mac model.

Use Target Disk Mode

If you can boot your computer (even if the screen is dead) put it into Target Disk Mode by booting it while simultaneously holding the T key.  The computers need to be connected by a Thunderbolt or FireWire cable.  The iMacs from this vintage (through late 2011) all came with a FireWire port.  Thunderbolt didn't come onto the scene until 2012.


FireWire Cable
Thunderbolt to Firewire Adapter - Only necessary if using a newer Mac with Thunderbolt ports


Once connected, you can boot the computers and copy the files over.

Remove the Hard Drive

Since your graphic card has died and they are quite expensive on the secondary market, it makes sense to simply extract the drive and connect it to a USB adapter to copy the files over.

The iMacs through late 2011 all came with the removable glass where you could simply use a couple of suction cups to pull the glass off.  The display panel was held in with 12 Torx screws.  Once removed, you have access to all of the components.  Ifixit has an excellent step-by-step guide to do this.


 



* Source:  iFixit.com. Photo credit: Sam Goldheart and licensed under CC BY-NC-SA 3.0.

Remove the HDD and connect it to another Mac with a USB to SATA Adapter with Power.  The drives in the iMac are 3.5" so they need a 12V supply - more than what a USB port can provide by itself.  If you have a spare 3.5" drive enclosure, that will work as well.

Again, once connected, you can copy your files over from the drive to your connected computer.