The Monte Carlo method is employed to study the conditions for nanoconductor formation on a DNA molecule in aqueous solution via electrostatic interactions between negatively charged groups of the matrix molecule and positively charged functionalized gold nanoparticles. A model is developed that explicitly takes into account low-molecular-mass counterions resulting from the dissociation of DNA molecule and surface functional groups of nanoparticles. It is shown that the explicit regard to the counterions is of great importance for investigating conditions of nanoaggregate formation. The model parameters are estimated at which continuous metal coatings are formed via self-assembly.