The elucidation of the structure of DNA began as an exercise in x-ray crystallography. If you shine a beam of x-rays through a crystal, it will act like a diffraction grating, and the points of constructive interference will show up as dark spots on photographic film. This had been used for decades to determine the structure of relatively simple molecules. A fairly sophisticated mathematical machinery had been developed that allowed the arrangement, distances, and intensity of the spots to be translated into the relative arrangement of the atoms in each unit cell of the crystal. After the war, many labs were using x-ray crystallography to determine the structure of the much more complex bio-molecules like protein and DNA. Rosalind Franklin in the laboratory of Maurice Wilkins at King's College London was producing high-quality x-ray crystallography of DNA, hoping to determine its structure. This was before digital computers were widely available, so grinding through all the computation to translate the positions of the dots on the photographic negative into the positions of atoms took forever.
Wilkins showed Watson a particularly high-quality photograph of the diffraction pattern produced by Franklin. Using the general structure of the diffraction pattern, knowledge about allowed bond lengths, and the biological constraints that required the number of 'A' bases to match the number of 'T' bases, and the number of 'G' bases to match the number of 'C' bases, Crick and Watson were able to assemble first cardboard, and then wire frame models of likely structures for DNA. This allowed them to skip almost all of the computations, and enabled them to scoop Franklin, Wilkes, and Linus Pauling.
So the discovery of the structure of DNA used x-ray crystallography, physical chemistry (figuring out reasonable bond lengths), some biological rules of thumb, and clever model making. Of course a huge amount of work in biology had been needed prior to this, to figure out that knowing the structure of DNA was worth anybody's time.