If you want to do GPU processing, your best bet is to get a GPU that is not driving display, so it can be put into TCC mode and used exclusively for compute. The general rule is that Tesla, Quadro and GeForce Titan cards can be put into this mode. So instead of upgrading your existing GPU, the better plan would be to keep that one for driving the display and get a new card for doing the GPU compute that will run alongside it.
The main concern with running a GPU for compute for long periods are power and heat. You need to have a power supply in the host that can provide adequate power to the GPU as well as other system components, and have enough cooling capacity to move the heat out of the chassis.
The Tesla line of cards is generally passively cooled and designed to go in a server enclosure which provides adequate cooling, so it's unsuitable for a workstation. You would probably want to select a card from the Quadro or GeForce Titan line. Most full size workstations I'm familar with should be able to do a display GPU and one additional compute GPU (2 full size PCI slots on the motherboard) -- but that's dependenent on the form factor of your workstation as well as power supply. Going to display card + two compute GPUs can be done in some workstation lines, and there are some deskside workstations that will support 4 GPUs doing compute -- but these are generally spec'd / ordered specifically.
There are plenty of vendors that support any of the above (and more), but of course in the end it comes down to what your budget is.
