Graded-index multimode optical fibers have recently attracted a renewed attention, thanks to the discovery of new nonlinear effects, such as Kerr beam self-cleaning. In essence, Kerr self-cleaning involves a flow of the propagating beam energy into the fundamental mode of the fiber, accompanied by a redistribution of the remaining energy among high-order modes. Increasing the fundamental mode energy leads to a significant improvement of the output beam quality. A standard method to determine beam quality is to measure the M2 parameter. However, since self-cleaning involves the nonlinear redistribution of energy among a large number of fiber modes, measuring a single beam quality parameter is not sufficient to characterize the effect. A properly informative approach requires performing the mode decomposition of the output beam. Mode decomposition permits to evaluate the energy distribution among all of the excited fiber modes, which enables investigations of nonlinear mode coupling processes at a qualitatively new level. In this work, we demonstrate an efficiency mode decomposition method based on holography, which is suitable for analyzing the self-cleaning effect. In a theoretical study, we describe the solution of the mode decomposition problem for the modes of the graded-index multimode fiber. In an experimental investigation, we demonstrate the decomposition of both low-power (speckled) and self-cleaned beams, involving more than 80 modes.