Should I go with ?
- HDMI
- DVI
- DisplayPort
DisplayPort. It's no contest.
"Drives three 4k monitors" as a "must have" makes it pretty easy to answer the rest of the question:
Intel GPUs can drive three displays, but I don't know of a single mobo that has three DP outputs; there aren't even many with two. Consumer (read: gamer) oriented AMD and nVidia cards typically have one DP output, and very few have two. So for three DP outputs, you're pretty much limited to professional graphics boards: nVidia's Quadro and NVS lines, AMD's Radeon Pro (or older FirePro) series, or something uncommon like a Matrox C680. My recommendation is an AMD FirePro W4100 or NVS 510, both of which are about $250.
If your mobo has two 16× PCIe slots, then it might seem like a cheap solution would be to buy two cards, each with dual DP outputs. But my experience with Xinerama back before cards had multiple (independent) outputs was that this setup was always finicky. Googling "how to set up three monitors in X" suggests that things haven't improved.