Monitor has almost no bearing on it, likewise for the computer. The things you have to be prepared for are the power supply (can it deliver the necessary wattage/amps?) and the size of the case (will a new card fit?). The monitor can then be considered (what size is the monitor? Do you need a high end card to drive it or is it smaller than 24" and just about any mainstream card will work?)
And of course, price.
Also, there's no DX12. Yet.