The Double Helix is an autobiographical narrative of the discovery of DNA structure, traditionally accredited to Watson and his partner Francis Crick. The book remains controversial for its unapologetic portrayal of rivalry, discrimination and competitiveness in the scientific community.
In general terms, there is use throughout the book of the scientific method, with which we are probably all familiar; hypothesis, experiment, results, conclusion and revision. The exact manner in which this is done is never prescribed; in fact at professional levels the entire process can be a bit of a scramble. For example, the same idea can be independently investigated, or forgotten, by unrelated individuals, such as Watson and Crick's competition with Linus Pauling. This isn't exactly a scientific approach; one might argue that if science was really the objective, Watson, Crick and Pauling would all be working together.
The two most prominent processes used in the discovery were x-ray diffraction imaging and models. Both processes are used simultaneously and informed by each other, and it was largely their willingness to combine data sets from their own research and research done by others that led Watson and Crick to the discovery.
X-ray diffraction is a process that uses X-rays to determine the atomic structure of an object. X-rays are just a high-energy form of electromagnetic radiation, like light. One of the properties that distinguishes different types of matter to each other is their interaction with EM radiation. In the case of x-ray diffraction, think of the sample as being like a disco ball, with a thin beam of light fired at it. As the disco ball rotates, the light is reflected in a regular pattern around the room. With enough careful measurement of those reflections, the structure of the disco ball could be determined, even if you couldn't actually see it. Likewise, the structure of a molecule can be determined, based on how the x-rays are reflected.
X-ray research was largely performed by Rosalind Franklin, who in some people's opinions did not receive the proper credit for her contributions to the DNA discovery, especially because she wasn't fully aware of or involved in Watson and Crick's use of her research in their own. Franklin's projects provided important data such as the fact that DNA is helical and antiparallel.
Models were important because the X-ray data couldn't really identify the exact identity of each individual atom, or its placement, nor could it offer sensible arrangements of those atoms in terms of chemical bonds. Physical models, sometimes as simple as cardboard, allowed Watson and Crick to "play" with structures until they found arrangements that seemed to fit the data and agreed with known chemical laws. One such (inaccurate) model placed the phosphates on the inside of the helix and the bases on the outside; this contradicted the observed interaction of DNA with water because the bases were known to be hydrophobic.