Seems like I've found a temporary solution for my problem. Managed to launch my game today by first manually starting "BEService.exe" and "BEService_x64.exe" in "E:\Program files\Steam\steamapps\common\DayZ\BattlEye" and then launching the game from "E:\Program files\Steam\steamapps\common\DayZ". If I try to launch it via Steam I still get the error every time so that's why I'm calling it a temporary solution. Would be nice to get this fixed soon since I usually get disconnected from every server after 15-20 minutes for some reason.
EDIT: Well seems like the game itself is working but the main problem is Battleye atm. The game crashes when I try to join any servers and opens cmd (http://i.imgur.com/W36gxiQ.png). I tried launching Battleye via cmd, like it tells me to do, but then I get the original error message (http://imgur.com/hl6Uu6h). I also tried downloading the BEClient.dll from their own website but no luck with that either.
EDIT2: I think I've succesfully narrowed down the problem to my Battleye service constantly crashing. I just tried starting it manually from task manager and I was able to log into a server and run around for about 10 seconds. After that game crashed and BEService was stopped in task manager. Though, if I start it and just let it run by itself (without starting DayZ) it doesn't stop.