I had a similar problem with a laptop and what I found is that it's possible with some laptops, and it's not with others. I ended up having to replace the screen on mine before I could re-install windows on it.
The issue is that some video card configurations in laptops don't enable HDMI output by default and need either the OS or the BIOS to enable them. My laptop wouldn't enable it with just the BIOS, and the Windows installer didn't enable it either. I could, on the other hand, install Linux on the machine using the external monitor.
A technique that might work for you is to plug in a USB mouse and keyboard and close the lid of the computer. If your laptop supports it, this can make the BIOS disable the laptop's KVM and use HDMI out instead. For details, see: http://www.tomsguide.com/faq/id-3694215/install-windows-laptop-broken-screen.html
There's another way that I thought about but never actually tried, so take this with a grain of salt: use bootable media with a boot loader that does enable the HDMI out on your machine (try GRUB, LILO, Syslinux, etc., find one that works for you), set it up to load your Windows installation media and run the Win install after the external monitor is working. This might work, or it might turn on HDMI out while the boot loader is active and turn it back off once the Win installation media is loaded.