Before you go updating the BIOS read below carefully.

BIOS updates for Sandybridge boards to add Ivybridge support have been known to be problematic if the correct steps are not followed exactly. This is not a problem if you are using an Ivybridge CPU as that would mean the potentially problematic update is already in place. Once that update has been performed all subsequent updates will behave as normal. If memory serves the update process is 2 fold and will continue after the initial flashing stage on the next boot. Removing the flash drive containing the update or powering off the system prior to the second phase completing will brick the BIOS. If you do choose to go ahead with an update and you are using a Sandybridge CPU you must keep this in mind. Use instant flash with the update saved to a fat32 formated flash drive and do not remove the drive or power off the system until it has successfully passed the post process and begun loading the OS. I would recommend allowing it to boot into the OS then restarting at least once from within the OS before powering off the system just to be safe.

Scary as all this may sound the reality is actually pretty simple, the update in question updates 2 elements of the boards functionality requiring a restart between the 2 stages, interrupting between the 2 stages is what causes the issue.

Now all this is moot if you are already using an Ivybridge CPU, in which case go ahead and update to v3.30 if you have not already. It is always wise after an update to allow the system to fully boot before removing the flash drive containing the update or shutting the system down however  Users who follow that basic precaution never encountered the issue I mentioned above.

What BIOS are you currently running? If it isn't 3.10 or greater then you may need to flash to 3.10 first and then to 3.30.

1. You have switched from Nvidia to AMD on the GPU, some threads suggest that not removing the old drivers correctly can cause a 62 post code. Try booting off the old GPU and using DDU (display driver uninstaller) from safe mode. Not sure how this could cause a post code error but I found the same info in a few different places so it's worth a shot. In all honesty though I would just try disconnecting the HDDs and see if it posts without one.

2. Have you tested the new GPU in another system or otherwise know for sure that it is working? You can try cleaning the contacts on the GPU and blow out the PCIe slot with some canned air. If the old GPU had been in the system for a long time it is possible some dust may be causing poor contact.

3. Make sure no peripherals are connected to the system, apparently some USB devices can cause the 62 post code in conjunction with certain GPUs.

The BIOS version should remain on 3.30 as it will have maximum compatibility for your setup.  I am afraid this list isn't as helpful as I would have liked 

It was working fine one day, then yesterday nothing

It could be a bad VBIOS flash that has bricked the card

The memory may have failed

The VRMs could be dead

The core could be cracked due to messing with the cooler

Something could be shorting on the PCB (fluid from water cooling, cat pee etc)

Blown output fuses/resistors, you would have to locate these and test them with a multi-meter.

The best place to start is to strip off the cooling unit and thoroughly examine the PCB with a magnifying glass looking for residue, burn marks, missing components and a cracked/chipped core.

Once you have done that and if everything looks to be in order you can clean the card carefully with isopropyl alcohol and an old soft toothbrush. Be sure you clean the contacts on the PCIe connector thoroughly. 

Now when you say there is no signal from the GPU, do you know if the GPU is detected by the OS? To test this you would need to run it in a system with an iGPU so you can boot from the onboard graphics with the 7870 connected to the system. If it is detecting in the OS then there may be hope. If not you can try and download the latest vBIOS for the GPU from it's manufacturer website and see if you can flash the vBIOS using the correct tools. If it was a bad flash that may revive the card.

There are other more drastic steps that can be taken but it is best to try all this first as the "last resort" I have in mind could just kill the GPU completely.

Indeed, for that price it is totally worth having some fun with it and trying to make it work again. Once he has done all the things suggested already I can suggest a few more things I have done on old GPUs to breathe life back into them in the past. 

I used to have a bin full of old, dead GPUs that I would pirate parts from (usually fans and coolers) and occasionally when I was bored would try and revive some of them. The most common cause of failures on higher end cards were failed attempts at "unlocking" them via vBIOS updates. In many cases flashing them back to stock revived them but the process can be tricky. The next most common was failed attempts at adding aftermarket cooling/water cooling where the user cracked/chipped the core. Sadly that is not correctable.

I also agree with Parsec and Wardog on the PSU issue, I would replace it and see if your 560 stops artifacting.

I personally suspect that the outputs are blown. If it was bad memory or a bad core typically the card will display something, usually flashing colors, garbled nonsense etc. There should be tiny fuses connecting to each of the display outputs but also a further array of fuses and circuitry that connects to all of those. If all the outputs are providing no signal then likely something is damaged at a point before the signal is split to all the outputs. Trace the circuits very carefully back from one of the outputs and find the point the signal is split. Examine the components in that area very carefully, you should find a blown fuse, resistor or IC. 

Interestingly, if you know any electronics gurus this problem should be easy for them to trace and correct, if it is only an output issue then it is no different than the circuitry found on a TV, radio or any other device that has inputs/outputs. I have a friend that usually does this kind of stuff for me for free but depending on where you live it may be cheaper and easier to give it to a professional to check the output circuitry.  

All the components following a circuit trace should provide resistance though so if you find one that does not that looks anything like these:

They should absolutely have a resistance rating on a test meter. If you find one that does not (it must look visually similar to these, the color can vary) you can try the pencil trick.

For the pencil trick you basically draw a line on the resistor/fuse in pencil that connects the 2 solder points. This should be conductive enough to get signal on the GPU if that is the only damaged component. The thicker the line the greater the conductivity. Generally any component soldered on to the PCB at each end like this should have a resistance rating. Now, once you identify the damaged component/components and tested with the pencil trick you really should replace the broken components with matching ones. There should be numbers on the components that will allow you to order them. If you do get this far you need to be careful to orient the replacements the right way or you will damage other components and probably brick the GPU completely.