What would 黑著臉 mean?
It means someone is not happy about things that are going on, it can be a disapprove to the decision other people made, anger about people's comment, behavior. 
黑 in this phrase describes a negative mood in one's face, as if someone is so mad at something, his/her heart begin beating faster, his/her face turns into a dark color than usual. Similarly you may see红着脸(flushed with fever or success or pride)， 绿着脸(bad stock market or unhappiness)，青着脸(angry or discontent about something). Other than color, there are also 板着脸，绷着脸，拉着脸，蒙着脸 etc.