In the present work, the near-field radiative heat transfer of a multilayered graphene system is investigated within the framework of the many-body theory. For the first time, the temperature distribution corresponding to the steady state of the system is investigated. Unique temperature steps are observed near both boundaries of the system, especially in the strong near-field regime. By utilizing the effective radiative thermal conductance, the thermal freedom of heat flux in different regions of the system is analyzed quantitatively, and the cause of various temperature distributions is explained accordingly. To characterize the heat transfer ability of the whole system, we evaluate the system with two heat transfer coefficients (HTC), transient heat transfer coefficient (THTC), and steady heat transfer coefficient (SHTC). A unique many-body enhancement is observed, which causes a red-shift of resonance peak corresponding to graphene surface plasmon polaritons. Furthermore, a three-body enhancement of SHTC emerges thanks to the relay effect and the complexity of the system. The regime of heat transport can be tuned by changing the chemical potentials of graphene and undergoes a transition from diffusive to quasi-ballistic transport in the strong near-field regime.