HDMI Adaptor Board - Blank Screen [not working]
Horrible signal integrity!

The TMDS channels (and TMDS Clock) are FAST as in multi GHz edge rates. You cannot route them without respecting that nature. 

The ground connections should be either thru or all commoned (Common is the way to go on a PCB because you need a ground plane on the board to define the pair impedance.  

On the PCB you need all the TMDS lines to be run over a ground plane with a controlled impedance, this usually means you want either a buried ground plane or a thin PCB to get the trace widths reasonable at the target impedance value. 

It is all in the HDMI electrical specifications, and is really not rocket science, but you need to think like an RF (and low band microwave) designer, and that has a somewhat undeserved reputation for being black magic.

A search on "Signal integrity" will bring up more then you ever wanted to know.