The spread of COVID-19 has sparked racism and hate on social media targeted towards Asian communities. However, little is known about how racial hate spreads during a pandemic and the role of counterspeech in mitigating this spread. In this work, we study the evolution and spread of anti-Asian hate speech through the lens of Twitter. We create COVID-HATE, the largest dataset of anti-Asian hate and counterspeech spanning 14 months, containing over 206 million tweets, and a social network with over 127 million nodes. By creating a novel hand-labeled dataset of 3,355 tweets, we train a text classifier to identify hate and counterspeech tweets that achieves an average macro-F1 score of 0.832. Using this dataset, we conduct longitudinal analysis of tweets and users. Analysis of the social network reveals that hateful and counterspeech users interact and engage extensively with one another, instead of living in isolated polarized communities. We find that nodes were highly likely to become hateful after being exposed to hateful content. Notably, counterspeech messages may discourage users from turning hateful, potentially suggesting a solution to curb hate on web and social media platforms. Data and code is at http://claws.cc.gatech.edu/covid.