I can guarantee you the following
1) Your power supply will not support the higher end GPUs because of
a)Not enough amps @ 12V
b)Probably doesn't even have PCIE power connectors
2) Your case is likely not sufficiently ventilated to accomodate such a card without said card throttling
If you are at the point where you are ready to do things like add a better GPU, you are at the point where you can consider building your entire computer yourself.
The most you can get into these prebuilt machines that are not geared towards gaming is usually a low power card that does not require PCIE power such as (most but not all) 750Ti's. Those are not powerful enough for a compelling VR experience anyways.
If you can afford the Oculus Rift, you can afford to fix your problem. The motherboard you have likely does not have standard mounting or use standard power supply connections. Find out if it is fully ATX compliant (do some googling). If it is you will need:
A new case (ie, Corsair 200R), a new power supply (ie, Corsair CX600). Everything else can be stuck in the new case. Add your GTX 970 or 980 or better yet 1070 or 1080 and you are good to go.
If the motherboard is NOT fully ATX compliant, you will need a new motherboard as well. You do not have K sku CPU so an H97 motherboard will be more than enough for you.
Be aware that if you do not have an ATX compliant motherboard, your CPU cooler may also not be compliant with the standard socket 115X mouting, and so would not work on a standard board. In this case you will need a new CPU cooler such as a coolermaster Hyper 212.
