In this paper, we study the entire radial solutions of the self-dual equations arising from the relativistic SU(3) Chern-Simons model proposed by Kao and Lee (Phys Rev D 50:6626-6632, 1994) and Dunne (Phys Lett B 345:452-457, 1995; Nuclear Phys B 433:333-348, 1995). Understanding the structure of entire radial solutions is one of the fundamental issues for the system of nonlinear equations. In this paper, we prove that any entire radial solutions must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. Even for radial solutions, this classification has remained an open problem for many years. As an application of this classification, we prove that the two components u and v have intersection at most finite times.