Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Durham to Moscow as you normally would.
The most popular route from Durham to Moscow is Raleigh-Durham - Moscow Sheremetyevo. You may also consider Raleigh-Durham - Moscow Domodedovo or Raleigh-Durham - Moscow Vnukovo.
This is defined as Raleigh-Durham Airport (RDU) and Moscow Vnukovo Airport (VKO). Codes like Moscow Vnukovo Airport are often used to just call airports uniquely. Moscow Domodedovo Airport (DME), Moscow Sheremetyevo Airport (SVO), Moscow Zhukovsky Airport (ZIA) are some additional IATA codes for airports related to your trip.