We propose a dual-band achromatic focusing metasurface based on polarization multiplexing and dispersion engineering. An anisotropic resonant phase meta-atom is designed to realize independent nonlinear phase manipulation along the orthogonal directions. Achromatic focusing metasurface and broadband reflectarray antenna are further constructed in the microwave region with a computer-assisted particle swarm optimization algorithm. The standard deviation of focus offset at 11-16 GHz (for x-polarization) and 18-24 GHz (for y-polarization) are compressed to 19.83% and 16.60% of the dispersive metasurface, respectively. The radiation gains of the reflectarray antenna increase by an average of 19.49 dB and 15.08 dB in the broadband region compared with the bare standard rectangle waveguides. Furthermore, such an achromatic metasurface can be utilized to realize different functions with polarization selectivity and applied to other frequency ranges, which holds great promise in integrated optics.