To acquire this quest, speak with Abraham Washington in Rivet City. He can be found inside the Capitol Preservation Society. Head to the National Archives and enter to find a woman named Sydney who was also sent by Abraham. She'll ask you to help in fighting off some super mutants, then she'll join you to look for the Declaration. Once you enter the room with the Declaration, a robot who thinks he helped sign the document approaches. With good speech or science skills you can immediately get past him through dialogue. Otherwise you might need to retrieve some ink from the Arlington Library so he can forge a copy to give you. Whether you have the real deal or the forgery, doesn't matter. Return it to Abraham to complete the quest.
See Stealing Independence for more specific info.