How was the composition of DNA studied before the knowledge of its structure?
In the early experiments performed by Erwin Chargaff and co-workers to determine the composition of DNA, leading to the deduction of the Chargaff's rules, the total DNA was extracted, chemically digested and the products were analysed using paper chromatography. The DNA was not broken down to the level of atoms which is certainly impossible using the standard chemical treatment procedures.

This is an excerpt from one of the earliest papers by Chargaff [1]:


  Two procedures were compared. In one, which will be referred to here
  as Procedure 1, the nitrogenous components were liberated and
  determined by the methods described previously from this laboratory
  (5, 8, 12, 13); i.e., the purines were set free by N H2S04 (1 hour,
  100o) and chromatographed in aqueous butanol-diethylene glycol
  (NH3 atmosphere); the pyrimidines were liberated with
  concentrated formic acid (2 hours, 175o) after removal of the purines
  as the hydrochlorides by treatment with methanolic HCl, and
  chromatographed in aqueous butanol.
  
  The second method, Procedure 2, is considerably simpler and permits the
  use of even smaller quantities of DNA (2 to 6 mg. for a complete
  analysis) than was possible heretofore; its use was briefly mentioned
  in a recent paper (14). It is based on the simultaneous liberation of
  all purines and pyrimidines by the hydrolysis of the DNA with
  concentrated formic acid and on their separation, on the same
  chromatogram, by means of ammoniacal butanol. When it is desired to
  establish complete balances, i.e. to express the molar distribution of
  individual nitrogenous constituents not only in terms of P present in
  the hydrolysate, but also in relation to the total N and P of the
  nucleic acid, about 6 mg. of DNA are used in the analysis. The sample
  (dried in a high vacuum at 60o for 3 hours) was heated in a sealed
  small bomb tube (about 220 × 6.5 mm.) with 0.8 cc. of concentrated
  formic acid (98 to 100 per cent) at 175o for 2 hours. The hydrolysate
  was transferred quantitatively to a 1 cc. volumetric flask and
  adjusted to volume by means of small portions of distilled water which
  served to rinse the hydrolysis vessel. Aliquots, measured with an
  accurate microburette in the usual manner (12), were employed for the
  calorimetric P determination (0.04 cc. of hydrolysate) and for the
  chromatographic separation (0.01 cc.). When very small quantities
  of DNA were to be analyzed, 2 mg. of the substance were heated with
  0.2 cc. of concentrated formic acid and the hydrolysate was, without volume adjustment, analyzed for P and for individual purines and
  pyrimidines. In this case, all computations were based on the ratio of
  moles of nitrogenous constituent to moles of phosphorus present in the
  hydrolysate (13, 14). The chromatographic and spectroscopic techniques
  were, in general, similar to those described previously (8, 12). The
  paper sheets (16 × 46 cm.), carrying five lanes 3 cm. wide, were left
  for 20 minutes in an ammonia atmosphere in order to neutralize the
  acid. The solvent used for the chromatographic separation consisted of
  a mixture of 6 volumes of n-butanol and 1 volume of 0.6 N aqueous
  ammonia. Two beakers, one containing the solvent mixture, the other
  0.6 N ammonia, were placed along- side each paper sheet at the bottom of the jar used for chromatography; the paper sheets were suspended,
  the solvent trough being left empty, and allowed to equilibrate with
  the solvent atmosphere for 1 hour in the closed vessel. Then the
  trough was filled with the solvent mixture and the separation
  permitted to proceed for 18 to 20 hours, at which time the solvent
  front had passed over the lower edge of the paper. After the papers
  had dried in air, the position of the separated components was marked
  under a Mineralight lamp or a similar ultraviolet lamp.4 The
  components, in the order of increasing distance from the starting
  point, were aligned as follows (the distance of thymine arbitrarily
  taken as 100) : guanine (26), cytosine (47), adenine (71), thymine.
  The quantitative determinations were carried out as described before
  (12).




 Chargaff, Erwin, et al. "The composition of the desoxyribonucleic acid of salmon sperm." Journal of Biological Chemistry 192.1 (1951): 223-230.