Melanoma is a skin cancer characterized by rapid growth and spread for which current therapies produce both resistance and increased risk of infection. To develop new anti-melanoma biocompatible species, the series of complexes Cu(N-N)(bzac)(X)⋅nH2O (N-N: 1,10-phenanthroline/2,2′-bipyridine, Hbzac: 1-phenyl-1,3-butanedione, X: NO3/ClO4, and n = 0, 1) was studied. Single-crystal X-ray diffraction revealed a mononuclear structure for all complexes. The ability of the complexes to scavenge or trap reactive oxygen species such as O2⋅− and HO⋅ was proved by EPR spectroscopy experiments. All complexes inhibited B16 murine melanoma cells in a dose-dependent and nanomolar range, but the complexes with 1,10-phenanthroline were more active. Moreover, comparative activity on B16 and healthy BJ cells revealed a therapeutic index of 1.27–2.24. Bioinformatic methods were used to calculate the drug-likeness, pharmacokinetic, pharmacogenomic, and pharmacodynamic profiles of the compounds. The results showed that all compounds exhibit drug-likeness features, as well as promising absorption, distribution, metabolism, and excretion (ADME) properties, and no toxicity. The pharmacodynamics results showed that the neutral species appear to be good candidates for antitumor molecular targets (Tyrosyl-DNA phosphodiesterase 1, DNA-(apurinic or apyrimidinic site) lyase or Kruppel-like factor 5). Furthermore, the pharmacogenomic results showed a good affinity of the copper(II) complexes for the human cytochrome. These results recommend complexes bearing 1,10-phenanthroline as good candidates for developing drugs to melanoma alternative treatment.