How do I fit a normal PCI-E x16 video card into a PCI-E x1 slot? What hardware would I need to buy to fit a normal PCI-E x16 card (In my case, a R9 280x) into a normal PCI-E x1 slot?
You need a PCI Express (PCI-e) x1 to x16 Riser adaptor cable like this one:
http://www.trademe.co.nz/computers/cables-adaptors/other/auction-670675802.htm

Bear in mind that some graphics cards draw a lot of power, and I've seen photos of cables that have overheated because they couldn't handle the load. The linked one above may have solved this issue by providing a separate power cable.