Non-orthogonal multiple access (NOMA) and multiple-input multiple-output (MIMO) are viewed as promising techniques to meet the demands for high spectrum efficiency and massive connectivity in future wireless communication. In this paper, a novel framework for realizing transmission in cell-free massive MIMO-NOMA system with deep integration between MIMO and NOMA is proposed. A new method is developed to divide users into different groups according to their equivalent path loss, then ``group-level successive interference cancellation'' (GSIC) is used to cancel inter-group interference when demodulating users. Based on the new framework, a rigorous closed-form expression of achievable sum rate in uplink cell-free massive MIMO-NOMA system is derived. Besides, a parallel iterative method is used to obtain the best power control scheme. It is shown by simulations that the proposed scheme can effectively reduce total power consumption and outperforms the orthogonal multiple access (OMA) transmission and the traditional SIC-NOMA schemes based on user clustering.