In this paper, we develop a new class of conservative characteristic finite difference methods for solving convection-dominated diffusion problems with fourth-order accuracy in both time and space. Specifically, the method of characteristics is utilized to handle the convection term, which allows for greater flexibility in the choice of time step sizes. To achieve high-order temporal accuracy, we propose characteristics-based optimal implicit strong stability preserving (SSP) Runge-Kutta methods implemented along the streamline. Furthermore, a conservative interpolation is employed to calculate values at the tracking points. By introducing diverse fourth-order approximation operators on the uniform Eulerian and irregular Lagrangian meshes, we can deal with the diffusion term with high accuracy while preserving the conservation property. The mass conservation for our proposed method is theoretically proved, and is verified through numerical experiments. Moreover, the numerical tests demonstrate that our scheme achieves temporal and spatial fourth-order accuracy and generates non-oscillatory solutions, even with large time step sizes.