Well This could be very easy .
You can use the following code
string sourceFolder = @"c:\update\files";
string outputFolder = @"D:\weeklybackup\"+Convert.ToString(DateTime.Today);
FileSystem.CopyDirectory( sourceFolder, outputFolder );
I always prefer This code over Directory.Move Method because FileSystem.CopyDirectory takes care of all folder,subfiloder and files.
Now after building / compling your code. Dont write a service. Simply add your exe to schedule tasks in your server. I hope u know what are schedule task and how they work.