From femtosecond bond vibrations to millisecond domain motions, the dynamics of biomolecules spans a wide range of time and length scales. This hierarchy of overlapping scales links the molecular and biophysical details to key aspects of their functionality. However, the span of scales combined with their intricate coupling rapidly drives atomic simulation methods to their limits, thereby often resulting in the need for coarse-graining techniques which cannot take full account of the biochemical details. To overcome this tradeoff, a graph-theoretical framework inspired by multiscale community detection methods and stochastic processes is here introduced for the analysis of protein and DNA structures. Using biophysical force fields, we propose a general mapping of the 3D atomic coordinates onto an energy-weighted network that includes the physico-chemical details of interatomic bonds and interactions. Making use of a dynamics-based approach for community detection on networks, optimal partitionings of the structure are identified which are biochemically relevant over different scales. The structural organisation of the biomolecule is shown to be recovered bottom-up over the entire range of chemical, biochemical and biologically meaningful scales, directly from the atomic information of the structure, and without any reparameterisation. This methodology is applied and discussed in five proteins and an ensemble of DNA quadruplexes. In each case, multiple conformations associated with different states of the biomolecule or stages of the underlying catalytic reaction are analysed. Experimental observations are shown to be correctly captured, including the functional domains, regions of the protein with coherent dynamics such as rigid clusters, and the spontaneous closure of some enzymes in the absence of substrate. A computational mutational analysis tool is also derived which identifies both known and new residues with a significant impact on ligand binding. In large multimeric structures, the methodology highlights patterns of long range communication taking place between subunits. In the highly dynamic and polymorphic DNA quadruplexes, key structural features for their physical stability and signatures of their unfolding pathway are identified in the static structure.