Measles is a highly contagious cause of febrile illness typically seen in young children. Recent years have witnessed the resurgence of measles cases in the United States. Prompt understanding of public perceptions of measles will allow public health agencies to respond appropriately promptly. We proposed a multi-task Convolutional Neural Network (MT-CNN) model to classify measles-related tweets in terms of three characteristics: Type of Message (6 subclasses), Emotion Expressed (6 subclasses), and Attitude towards Vaccination (3 subclasses). A gold standard corpus that contains 2,997 tweets with annotation in these dimensions was manually curated. A variety of conventional machine learning and deep learning models were evaluated as baseline models. The MT-CNN model performed better than other baseline conventional machine learning and the signal-task CNN models, and was then applied to predict unlabeled measles-related Twitter discussions that were crawled from 2007 to 2019, and the trends of public perceptions were analyzed along three dimensions.