An automatic aberration compensation method is proposed by combing phase fitting and deep learning in digital holographic microscopy and applications to dynamic observation of living cell morphology. Firstly, in the holography recording plane, the preliminary aberration compensation is implemented by using a digital phase mask, whose quadratic polynomial fitting computation is based on the extraction of reconstructed phase value along central cross line profiles in the field of view. Then, in the holographic imaging plane, the final aberration compensation is completed by using higher-order polynomial digital phase mask that computed with the phase data in the object free region, which is defined by the convolutional neural network. Thus, the phase image of object is correctly and accurately reconstructed. Thanks to the preliminary aberration compensation in the holography recording plane, the complexity of reconstruction in the holographic imaging plane is effectively reduced before the convolutional neural network training. Therefore, a stable deep learning model for phase image segmentation can be obtained on basis of limited data set and the compensation of phase aberration can be achieved without any manual intervention. The experiments are demonstrated by observing the several kinds of living cells, which have different morphological characteristics, with the off-axis digital holographic microscopy. Furthermore, the proposed method is applied to screen the drug resistance of endometrial cancer cells. These experimental results verify the proposed method and show that it can be used to dynamic microscopic observation in biological cell research.