Another thing to take into consideration when deciding between these two, is whether you will want a microphone. If yes, then X2HR are easier to do that with a V-Moda boompro, as they have a detachable 3.5mm cable.
You can add a microphone to the DT770, but you'd need a Modmic, which has it's own cable; and with the DT770 not having a removable cable, then you've got two cables to content with. That does frustrate some people, so the X2HR is a better choice in that regard.
As already mentioned; DT770 are harder drive, especially if looking at the 250 Ohm version. 80 Ohm version would be the better choice for use with onboard audio or anything without a dedicated headphone amp.
The HyperX Cloud II, has a USB dongle which serves as a sound card and has surround sound processing, if you want to call it that. By using the above headphones with onboard audio, you will lose that, unless you use something to replace the surround sound processing. Whether it be software or a sound card.
Windows 10 has a trial for Dolby Atmos for headphones and also has it's own free to use Sonic surround sound feature. I forget the exact name. You can use either in place of a sound card, but neither of those are as good as Creative's SBX prostudio. You can buy that software from Creative's site, but it is £30 and not particularly good value when they have B grade SoundblasterX G6 USB sound cards come up from time to time for £70. G6 for £70 would be a really good buy.