Good news for fans of Nuclear Winter! It appears any achievements that can be acquired in Adventure Mode such as getting kills, hacking terminals, etc., can all be attained in this mode!
In Nuclear Winter (battle royale) you only have to down your opponent for it to count as a kill as my achievement popped as I downed someone! I would recommend fully automatic weapons as people tend to not stand still in this mode and spraying usually helps. Also having a team or other squadmates can benefit you as they can call out enemy positions. If you are desperate for kills, tend to spawn in heavily popular areas as named cities and locations as most of the lobby at that time does not have much armor or powerful weapons.