In the long run, no, you can not be a double agent. However, you can be a double agent early on in the questline. First, you complete either Joining the Legion or Joining the Stormcloaks. Then, you do the quest The Jagged Crown. However, once you get the crown, do not take it to the leader of your side. Take it to the leader of the opposite side. This will let you join their side for the rest of the questline. Beyond that, there is no way to be a double agent in the Civil War questline.