We first hear about Judge Brack as the man who helped Aunt Julie arrange to put up the money for a security deposit. Then we learn that he helped George with the finances involved in buying the "dream" house. When he first comes over to the Tesmans, he’s bringing news with regard to George getting the post as professor, which he promises will still go through. So from the get-go, it’s clear that this is a guy who makes things happen. He’s the one to know. He’s got power, sure, but more importantly, he’s a man of action.
His willingness to act is one of a slew of similarities which Brack shares with Hedda. He’s certainly as smart as she is, which explains that razor-sharp banter back and forth between them (see "Style" for more). He’s intuitive enough to pick up on information that isn’t explicit; just as Hedda recognizes Thea’s relationship with Eilert before it’s admitted, so the Judge is able to guess at Hedda’s past relationship with the same man. Like Hedda, he feels out a situation for information before revealing his hand – you can see a great example of this in Act IV when he arrives with news of Eilert’s "suicide." We note in Hedda’s "Character Analysis" that her greatest asset is her ability to make others confess – and Brack is the only man able to turn this table on her. She ends up confessing to him. In fact, Brack is the only character that Hedda is ever honest with. (Check out the passage where she confesses the trick she played on Aunt Julie about her hat. She then says, "These things come over me, just like that, suddenly. And I can’t hold back." It’s unlikely that she’s faking here; this is Hedda without the manipulative cover.)
The Judge also shares Hedda’s dislike for traditional Victorian values. He seems to have no interest in marriage (she admits that she "never really held out any hopes for [him]" the previous summer), and expresses a penchant for "backways," which he finds "piquant." (He means that secrets and intrigue are more stimulating to the mind than what’s above the table.) It’s also worth noting that, at the end of Hedda Gabler, the Judge becomes the first character to ever have power over Hedda – to "hold [her] destiny" in his hands, as she so desired to do to Eilert.
Brack is in many ways the male Hedda. She should love this guy, right? Not so much. That probably has something to do with the fact that he tries to blackmail her into having an affair with him. But the resentment actually starts long before the coercion in Act IV. Hedda may banter with Brack, but she’s certainly not committing to anything. (When she tells Eilert that she won’t be unfaithful to George, we know that she’s flirting with the Judge, but with no serious intentions.) She also calls Brack "that disgusting Judge" when conversing with Mrs. Elvsted and Eilert, later condemning him to his face as being "a specialist," just like George. She probably means that Brack, like George, is obsessed with one goal and is therefore ultimately a bore. It’s just that Brack’s specialty is trying to sleep with Hedda, not the domestic handicrafts of Brabant in the Middle Ages, so he holds her attention for a little bit longer.