The authors have developed the characteristic Galerkin finite element method with marker integration as a novel Eulerian approach for large deformation problems of elasto-plasticity. In this paper, we apply this approach to the large deformation problem of incompressible hyperelasticity. To evaluate stresses in hyperelasticity properly, the total deformation gradient should be updated instead of stress integration for rate type constitutive models. In this paper, we employ a multiplicative update algorithm of deformation gradient developed for the updated Lagrange formulation of hyperelasticity.