Dell has tested following Graphics Cards with Dell XPS 8300 system:
Theoretically speaking you should be able to install the AMD Radeon 7870 card on the system, but you might need to upgrade the power supply to 500W or more to support it. However, Dell has not tested this card on this system. Hope this helps.
You may refer to the Service Manual of this system for instructions for replacing the components. Please follow the safety instructions.
Please reply if you have any questions.
So chances are it would work? Would any power supply work too?
The card should work with this system. I did some research online and found this link where few users have successfully installed the card and it is working fine.
Hope this helps. Please reply if you have any questions.