“Hamilton,” a hip-hop influenced musical about Alexander Hamilton at New York’s Public Theater, has quickly become the buzziest show of the spring, with a sold-out run off-Broadway, ecstatic tweets from tastemakers and a likely Broadway transfer—all before it’s even opened.

The show, written and composed by Lin-Manuel Miranda, who also plays Hamilton, is based on the biography of the founding father by Ron Chernow. The show began previews at the Public Theater on Jan. 20 and officially opens on Feb. 17 and closes May 3. It...