Viruses have evolved specialized mechanisms to efficiently transport nucleic acids and other biomolecules into specific host cells. They achieve this by performing a coordinated series of complex functions, resulting in delivery that is far more efficient than existing synthetic delivery mechanisms. Inspired by these natural systems, we describe a process for synthesizing chemically defined molecular constructs that likewise achieve targeted delivery through a series of coordinated functions. We employ an efficient "click chemistry" technique to synthesize aptamer-polymer hybrids (APHs), coupling cell-targeting aptamers to block copolymers that secure a therapeutic payload in an inactive state. Upon recognizing the targeted cell-surface marker, the APH enters the host cell via endocytosis, at which point the payload is triggered to be released into the cytoplasm. After visualizing this process with coumarin dye, we demonstrate targeted killing of tumor cells with doxorubicin. Importantly, this process can be generalized to yield APHs that specifically target different surface markers.