The paper presents the cellular automaton (CA) model for tracking the development of dendritic structure in non-equilibrium solidification conditions of binary alloy. Thermal, diffusion and surface phenomena have been included in the mathematical description of solidification. The methodology for calculating growth velocity of the liquid-solid interface based on solute balance, considering the distribution of the alloy component in the neighborhood of moving interface has been proposed. The influence of solidification front curvature on the equilibrium temperature was determined by applying the Gibbs Thomson approach. Solute and heat transfer equations were solved using the finite difference method assuming periodic boundary conditions and Newton cooling boundary condition at the edges of the system. The solutal field in the calculation domain was obtained separately for solid and liquid phase. Numerical simulations were carried out for the Al-4 wt.% Cu alloy at two cooling rates 15 K/s and 50 K/s. Microstructure images generated on the basis of calculations were compared with actual structures of castings. It was found that the results of the calculations are agreement in qualitative terms with the results of experimental research. The developed model can reproduce many morphological features of the dendritic structure and in particular: generating dendritic front and primary arms, creating, extension and coarsening of secondary branches, interface inhibition, branch fusion, considering the coupled motion and growth interaction of crystals.