Echocardiography (echo) is an indispensable tool in a cardiologist's diagnostic armamentarium. To date, almost all echocardiographic parameters require time-consuming manual labeling and measurements by an experienced echocardiographer and exhibit significant variability, owing to the noisy and artifact-laden nature of echo images. For example, mitral inflow (MI) Doppler is used to assess left ventricular (LV) diastolic function, which is of paramount clinical importance to distinguish between different cardiac diseases. In the current work we present a fully automated workflow which leverages deep learning to a) label MI Doppler images acquired in an echo study, b) detect the envelope of MI Doppler signal, c) extract early and late filing (E- and A-wave) flow velocities and E-wave deceleration time from the envelope. We trained a variety of convolutional neural networks (CNN) models on 5544 images of 140 patients for predicting 24 image classes including MI Doppler images and obtained overall accuracy of 0.97 on 1737 images of 40 patients. Automated E and A wave velocity showed excellent correlation (Pearson R 0.99 and 0.98 respectively) and Bland-Altman agreement (mean difference 0.06 and 0.05 m/s respectively and SD 0.03 for both) with the operator measurements. Deceleration time also showed good but lower correlation (Pearson R 0.82) and Bland-Altman agreement (mean difference: 34.1ms, SD: 30.9ms). These results demonstrate feasibility of Doppler echocardiography measurement automation and the promise of a fully automated echocardiography measurement package.