We investigate the three-body Coulomb breakup of a two-neutron halo nucleus, He-6. Based on the alpha + n + n model, the three-body scattering states of He-6 are described by using the combined methods of the complex scaling and the Lippmann-Schwinger equation. We calculate the breakup cross section, the two-dimensional energy distributions, and the invariant mass spectra for the E1 transition of He-6. We discuss the relations between the structures in these strengths and the n-n and alpha-n correlations of He-6. It is found that the He-5 resonance in the final states contributes to make a low-energy enhancement of the strength. The n-n final-state interaction also contributes to enhance the strength globally. However, the ground-state correlations of He-6, such as a dineutron, are difficult to recognize in the strength because of the dominant effect of the final-state interaction.