AFAIK DVI does not carry audio in any way shape or form. The Xbox One, however, does have an optical out. Something like this would carry the audio to your monitor, provided it has a 3.5mm input port http://www.amazon.co.uk/TOSlink-Digital-Toslink-Optical... You could then use the 3.5mm output port (again on the monitor if it has one, check the manual!) to carry audio to speakers or a headset, provided either use a 3.5mm jack. Alternatively, buy a headset that uses optical audio. I use Turtle Beach XP500s, had them for a while now, and although expensive, do the job extremely well. The only downside is that, although this headset is compatible with Xbox 360 chat, it is only compatible with chat on xbox one using an additional cable from TB and a stereo to chat adapter from MS. You could, however, just use the kinect microphone for chat, it does the job
EDIT: Apparently that cable won't work properly unless your monitor has a SPDIF 3.5mm in or something, this isn't my strongest subject so apologies for that. A note on the XP500s though, the wireless audio thing it uses (name escapes me) has a 3.5mm jack you could use if you wanted for external speakers for when you don't want to use the headset