"The night was black and starry with a waning moon ..."
from a history of the 57th Massachusetts Veteran Volunteers
The IX Corps moved from their lines at Cold Harbor on the night of June 12th, marching to the James River. Today's digital tools let us check the night sky for this movement.
One perpetual calendar gives moon phases for Richmond -- the moon rose at 12:23 in the afternoon of the 12th and set twelve hours later at 12:26 am on the 13th. The moon was at first quarter, waxing toward a full moon a week later on the 19th.
If the night was "black and starry" we are led to think there wasn't a great deal of cloud cover. But a "waning moon"? No, it was half full and setting in the west while the men were moving east.
Another wonderful tool is Starry Night from Simulation Curriculum Corp. This allows us to view the sky from any point on (or even off) Earth. A composite image from that program for the sky looking west is shown below (moon at 8, 9, 10 and 11 pm):
From Robert Krick we have Civil War Weather in Virginia and can find that the temperatures in Washington, D.C. (Georgetown), ranged from the low 60's at night to 75 in the day for the 12th and 13th. In notes he also reports that the weather around Richmond was cooler than normal (cold and cloudy, clear and cool on those days).