We describe the synthesis and characterization of a novel bioconjugate, consisting of an octaarginine cell-penetrating peptide and a highly DNA-affine doxorubicin dimer. The linkage between the two components is composed of a cleavable disulfide bond, which enables the efficient intracellular delivery of the cytotoxic payload within the reductive environment of the cytosol, mediated through glutathione. To determine the DNA-binding affinity of the dimeric drug molecule, microscale thermophoresis was applied. This is the first utilization of this method to assess the binding interactions of an anthracycline drug with nucleic acids. The cytotoxic effect of the peptide-drug conjugate, studied with drug-sensitive and doxorubicin-resistant cancer cells, demonstrates that the bioconjugate can successfully overcome drug resistance in neuroblastoma cells.