Online hate speech, particularly over microblogging platforms like Twitter, has emerged as arguably the most severe issue of the past decade. Several countries have reported a steep rise in hate crimes infuriated by malicious hate campaigns. While the detection of hate speech is one of the emerging research areas, the generation and spread of topic-dependent hate in the information network remain under-explored. In this work, we focus on exploring user behaviour, which triggers the genesis of hate speech on Twitter and how it diffuses via retweets. We crawl a large-scale dataset of tweets, retweets, user activity history, and follower networks, comprising over 161 million tweets from more than $41$ million unique users. We also collect over 600k contemporary news articles published online. We characterize different signals of information that govern these dynamics. Our analyses differentiate the diffusion dynamics in the presence of hate from usual information diffusion. This motivates us to formulate the modelling problem in a topic-aware setting with real-world knowledge. For predicting the initiation of hate speech for any given hashtag, we propose multiple feature-rich models, with the best performing one achieving a macro F1 score of 0.65. Meanwhile, to predict the retweet dynamics on Twitter, we propose RETINA, a novel neural architecture that incorporates exogenous influence using scaled dot-product attention. RETINA achieves a macro F1-score of 0.85, outperforming multiple state-of-the-art models. Our analysis reveals the superlative power of RETINA to predict the retweet dynamics of hateful content compared to the existing diffusion models.