The engineering and manipulation of delocalized molecular exciton states is a key component for artificial biomimetic light harvesting complexes as well as alternative circuitry platforms based on exciton propagation. Here we examine the consequences of strong electronic coupling in cyanine homodimers on DNA duplex scaffolds. The most closely spaced dyes, attached to positions directly across the double-helix from one another, exhibit pronounced Davydov splitting due to strong electronic coupling. We demonstrate that the DNA scaffold is sufficiently robust to support observation of the transition from the lowest energy (J-like) one-exciton state to the nonlocal two-exciton state, where each cyanine dye is in the excited state. This transition proceeds via sequential photon absorption and persists for the lifetime of the exciton, establishing this as a controlled method for creating two-exciton states. Our observations suggest that DNA-organized dye networks have potential as platforms for molecular logic gates and entangled photon emission based on delocalized two-exciton states.