You may end up running into all kinds of signal integrity issues - making cables/connectors for high speed digital signaling is not an easy feat. The best approach to take (if round/compact connector is absolutely required) may be like the following:
Feed DVI signal into an DVI to Ethernet cable adapter (like this one: http://www.iogear.com/product/GVE200/); those come in pairs, so on the other side of the link you end up with a normal DVI connector again.
Use any high quality ethernet cabling solution to make the connectors (the cables can be purchased).
If you're not satisfied with rectangular Rj45 ethernet connectors, you've got 2 options:
Sealed ethernet connectors (those are round on the outside: http://www.te.com/catalog/feat/en/c/20122?BML=10576,17678)
M12 industrial ethernet connectors (those are designed to be round: http://www.te.com/catalog/feat/en/s/29686?BML=10576,17678http://www.te.com/catalog/feat/en/s/29686?BML=10576,17678; example of a panel connector: http://www.sineco.com/P29930-Panel-Mount-Female-M12-Receptacle-to-RJ45-with-Belden_p_971.html)
With a bit of luck theThe above will require no soldering whatsoever.