It doesn't work "with" DX12, it replaces it. A game or piece of software will use either one or the other (both can co-exist in a system if that's what you meant and software can support multiple renderers (eg a game with a toggle between DX and OpenGL in the graphics settings)). But only one graphics API can be used at a time.
DX12 is a step in the right direction as it also..umm "borrowed" features from Mantle. Vulkan aims to be superior than DirectX though. And most importantly, for a lot of the players involved, aims to "decouple" gaming from windows. This is why Valve is pushing it hard, due to SteamOS/Steamboxes being Linux based. Nintendo is also behind it 100%.
As far as performance goes, both Vulkan and DX12 aim to eliminate the CPU bottleneck by allowing an unprecedented number of draw calls. Vulkan goes even further by inheriting all the best parts of Mantle. Considering DX12 is Win10 only and Vulkan can be run on any Windows version and other OS as well, adoption rate for Vulkan by developers is a much safer bet. Remember the raise in DX11 adoption only happened when MS brought it to Win7 as well. And this is the reason why AMD's Mantle failed in the first place. While it did not limit itself to Windows versions, it did run only on AMD cards. Developers want bigger audience, no one will spend resources to adopt a technology that's being utilized only by a small percentage of the userbase. Unless you're being paid to do so or have software engineers on loan, working on it, without any cost for you
😉Vulkan though runs everywhere and on everything. It's supported by both Nvidia and AMD and I'm quite sure Intel will also jump onboard for their IGP. My personal prediction is that we'll start seeing titles utilizing Vulkan by fall 2016.
