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I had the same issue. I got a black screen and then nothing.
Yesterday I looked at the system requirements and felt kinda stupid. My graphics card doesn't support DX11. But I still got it running. It's unplayable but at least it did launch. Here is what I did.
1. I installed the game (I downloaded it via gog downloader but at this point I'm not sure if it really matters).
2. I downloaded and installed the Windows SDK.
3. I launched dxcpl. Added the witcher3.exe to the list and checked the Box "Force WARP".
4. Launched the game with the witcher3.exe in the game directory.
Guess I have to get a new graphics card to actually play the game.