A series of homoleptic bis(terpyridine)copper(II) complexes of the type [Cu(L1–5)2]Cl2 (1–5), where L1–5 = 4′-(4-substituted)-2,2′:6′,2′′-terpyridines, have been synthesized and characterized. The molecular structure of complex 2 was confirmed by the single crystal XRD technique, and the geometry of the complexes is best described as distorted octahedral. Structural parameters from the crystallographic and DFT studies are in good agreement with each other. The small HOMO–LUMO energy gap supports bioefficacy of the complexes. DNA binding studies show high intrinsic binding constant values 1.53 ± 0.15, 1.62 ± 0.08 and 3.09 ± 0.12 × 105 M−1 for complexes 1, 2 and 3, respectively, with intercalative mode of binding to CT-DNA. The binding results were further supported by molecular docking studies. The experimental results indicate that the interaction between the complexes and BSA protein involves a static quenching mechanism. The molecular docking studies with c-Met tyrosine kinase receptors show hydrophobic and π–π interactions. All the complexes bring about hydroxyl radical mediated DNA cleavage in the presence of H2O2. In vitro cytotoxicities of the complexes (1–3) were tested against three cancerous cell lines, namely human breast adenocarcinoma (MCF-7), epithelioma (Hep-2) and cervical (HeLa) cell lines, and one non-tumorigenic human dermal fibroblast (NHDF) cell line by MTT reduction assay. The morphological assessment data obtained using Hoechst 33258 staining revealed that complex 3 induces apoptosis much more effectively than the other complexes.