The electronic structure and transport property of a carbon chain between two graphene nanoribbon leads are studied using an ab initio tight-binding (TB) model and Landauer's formalism combined with a non-equilibrium Green's function. The TB Hamiltonian and overlap matrices are extracted from first-principles density functional calculations through the quasi-atomic minimal basis orbital scheme. The accuracy of the TB model is demonstrated by comparing the electronic structure from the TB model with that from first-principles density functional theory. The results of electronic transport on a carbon atomic chain connected to armchair and zigzag graphene ribbon leads, such as different transport characters near the Fermi level and at most one quantized conductance, reveal the effect of the electronic structure of the leads and the scattering from the atomic chain. In addition, bond length alternation and an interesting transmission resonance are observed in the atomic chain connected to zigzag graphene ribbon leads. Our approach provides a promising route to quantitative investigation of both the electronic structure and transport property of large systems.