What specs must a graphics card have to power a 5K monitor?

You need a card with two DisplayPort 1.2 outputs. DVI connections are not used by this monitor.

Edit: this monitor connects with two DP cables and each cable appears as a single logical screen (there is no MST involved). So there is no requirement for the card to support more than two logical screens (which can be problematic for some Nvidia cards). Any card with two DP1.2 will be fine, therefore.

Many go on about how current video cards don't have the raw power for 5k, but unless you are gaming this is a non-issue. (Or unless you have other specialized 3D requirements, but if so you know what they are.) For ordinary use (which includes programming, web browsing, productivity applications, mail and so on) there is absolutely no performance issue with any recent video card. (I drive two UP2715Ks from one Nvidia NVS 510 card, which doesn't have a high-powered GPU.)

The problems are not in the number of rendering pipelines, GPU clock speed, etc. but just in whether the card has the right DisplayPort 1.2 outputs - and, perhaps, a smart enough driver to tile together two outputs into one desktop. If it has them, it will be adequate in other respects too.

Photo editing does need more power for higher resolutions, but this depends more on the resolution of the input file than on the monitor you're using (and in any case is usually CPU-bound not GPU-bound).

Dell says "Supported cards include: NVidia Kx000, NVidia x200 series, and AMD Wx100 series.". These are all professional cards and so cost more than a GeForce, without necessarily better performance, but the K2000 and K2200 are reasonably inexpensive. The NVS 510 also works (my experience). The same Windows driver from Nvidia handles all of these NVS and Quadro cards. (The NVS 310 is tested working but only at 30Hz refresh.)

Nowadays, though, the prices of the low-end Quadro cards have fallen so it may not be worth bothering with the slower NVS ones. I'm currently using Quadro M2000s on one machine and a Quadro P600 on another.

Although the NVS 510 card works perfectly to drive one or even two UP2715Ks at full resolution, I had the experience that Nvidia's support desk denied that any cards supported 5k -- even though support for it was mentioned in the release notes. That was in 2015 or so and has surely changed now. But it underlines the general point that what works and what Nvidia (or any company) calls 'supported' are not the same thing. You may also get one part of a company disagreeing with another part about what's supported.

(Slightly off topic: Note that if you want to drive 4k displays not 5k ones, the number of them you can drive from a Nvidia card usually depends on whether the display uses MST or not. A card like the NVS 510 with four outputs can drive four displays, but only two if each display uses MST to appear as two logical screens. This applies to some older 4k monitors like the Dell UP2414Q.)

You need to have two DisplayPort 1.2 connections in order to run at 5120x2880 with 60Hz refresh rate.

A GeForce GTX 960 is currently the cheapest way to get proper support for 5K on this Dell monitor. As mentioned, as long as you only want to use it for everyday work (CUDA acceleration in certain Adobe applications such as After Effects or Premiere may want some extra power), you'll be fine for GPU power.