Graphics card failing
If my graphics card fails can I hook up another monitor to the iMac and transfer data to another hard drive?
iMac, iOS 10.2.1, model late 2012 27"
No, if your GPU is shot hooking up an external display won't help at all. Your best bet is to use your Time Machine backup or bootable clone backup. However if you don't have backup you have options, first if you have access to another Mac connect via Target Disk Mode as explained in Use target disk mode to share files between two Mac computers - Apple Support. If you don't have access to another Mac then the next step is to remove the HD from the offending Mac and put into a high quality enclosure such as an OWC Mercury Elite Pro. If you don't want to to do that and you have no backup then consider your data gone gone gone!