That PSU looks a little wimpy to do the job, a 650W would be my recommendation, that could be the cause of your issue. I have a similar motherboard to you (Asus Z170a Pro gaming Carbon) and same CPU along with Windows 10 X64 and 2x 480 cards, 16GB of RAM but I am using a 1000W PSU.
Agree with Pete.
Also this could be a faulty GPU. Have you tried installing the 480 into a different PCI-E slot and then installing the drivers?
If you can try the 480 in a different system that might narrow down the problem.
I have this exact same issue...
I originally had a Sapphire RX480 Reference, that was suffering from this problem. (Black screen during and after driver install)
I put it down to the sapphire card being bad and sent it back.
Today I received a brand new XFX RX480 GTR (decided to upgrade if I had to send it back as I didn't want another Sapphire card)
And boom, exactly the same issue...
Done everything I can think to try, different versions of driver, minimal driver installation, allowed windows to install the driver itself using Windows update, tried a brand new fresh install of Windows 10 64-Bit on a different hard drive, tried both BIOS settings on the switch, nothing has worked! (Each time cleaning the registry and wiping all graphics drivers in safe mode when it didn't work) I have also flashed the BIOS to the latest version on XFX's website for the 480 to see if this would help, it didn't!
I have a 650 watt power supply, which should be more than enough to power the 150W quoted for the XFX 480.
I'm an experienced system builder and I have never had this much of a problem with a card before, let alone 2 separate cards!
You're not alone Karachokid !
This thread might help you: To people who have instant Black Screen or Freeze on installing ANY AMD driver
But as far as I can see, there's no fix that anyone has come across yet...