We consider the propagation of optical beams in an inhomogeneous nonlinear waveguide whose refractive index exhibits a dual power law dependence on the electric field amplitude. Self-similar evolution of the beams is studied within the framework of a generalized nonlinear Schrodinger equation with distributed diffraction, dual power-law nonlinearity, and gain or loss. The propagation properties and formation conditions of self-similar solitons in such a waveguide is studied by means of an improved homogeneous balance principle and an extended hyperbolic function method. A variety of new self-similar bright solitons are found and their characteristics are explored in terms of the inhomogeneous material parameters. A class of self-similar kink solitons is also identified, featuring another type of the self-similar evolution in the nonlinear waveguide. Dynamical behaviors of these self-similar structures are studied for a periodic distributed amplification system. It is shown that the parameters associated with the group velocity enable one to control the self-similar wave structure and dynamical behavior more efficiently, while the gain parameter affects just the evolution of the self-similar soliton's peak. Unlike the case of homogeneous dual power-law materials, soliton structures in inhomogeneous systems show novel and interesting features due to their distributed parameters which allow effective control of their dynamics.