Is your DP-HDMI adapter an "Active" or "Passive" adapter?
IF passive try using an Active DP-HDMI adapter and see if you get video output to the monitor. Also make sure your monitor settings are correct as far a Resolution and Video input.
Have you tried a different HDMI cable just to eliminate a bad cable?
Try connecting a Monitor with a DisplayPort Input and see if you still get the black video output. Thus eliminating the adapter as a possible reason.
Does Windows Device Manager show any errors or Yellow Exclamation marks?
How do you have Windows Display configured for both Monitors? Extended or Duplicate? Recommended Resolution?