This was the first time we've heard AMD's Linux team talk about Vulkan... They are working on a Vulkan Linux driver and have one prototyped, but initially it will be closed-source and the open-sourced later... They also mention that OpenCL support will initially be closed up and then opened later. They already have some basic OpenCL open-source support via the Clover Gallium3D driver, but I imagine they're referring to OpenCL 2.1+ support with SPIR-V alongside Vulkan or the OpenCL Catalyst code.
Their Vulkan driver is in user-space communicating with libdrm that in turn is interfacing with the AMDGPU kernel driver. The slides don't mention whether they intend to support Vulkan with the current Radeon DRM driver for HD 7000 through Rx 300 (non-Tonga/Carrizo/Fiji) GPUs but only reference the AMDGPU kernel driver just for the very newest AMD GPUs like the R9 Fury and Carrizo APUs. I'm sure that will be answered soon in our forums by Alex Deucher or John Bridgman.
In the future they intend to focus more on open-source than closed-source.
Closed-source is still relied on for workstation features, OpenGL 4.5, and OpenCL.
Basic support for their closed-source (Catalyst) OpenGL and OpenCL support riding atop the rest of the AMDGPU driver stack is complete. Their initial Vulkan driver is using DRI3.