In this paper, Hirota's bilinear method is extended to a new KdV hierarchy with variable coefficients. As a result, one-soliton solution, two-soliton solution and three-soliton solutions are obtained, from which the uniform formula of N-soliton solution is derived. Thanks to the arbitrariness of the included functions, these obtained solutions possess rich local structural features like the ridge soliton and the concave column soliton. It is shown that the Hirota's bilinear method can be used for constructing multi-soliton solutions of some other hierarchies of nonlinear partial differential equations with variable coefficients.