A prominent and menacing lawyer to the rich and powerful, Tulkinghorn sniffs out Lady Dedlock's secret. His pursuit of the truth about her past, without any concern about the enemies he leaves in his wake, ends in his murder.
Think of Tulkinghorn as the Terminator. Not because he's all muscles, but because he is just completely inexorable. (Hey, save that word for the SAT – it means unstoppable and relentless.) If Lady Dedlock gets an A in not displaying any emotions, then Tulkinghorn gets an A+. When he realizes there's a secret to be dug out from Lady Dedlock's past, he uses every trick in the book (bribery, blackmail, abduction) to get to the bottom of things. And all that time, he never raises his voice, never walks faster than a slow glide, and never lets a hair on his head fall out of place, especially when Lady Dedlock starts clearly being terrified of him.
But now for the more interesting question – why does Tulkinghorn do what he does? After all, he's making a good living keeping Sir Dedlock lawyered up. Why would he sabotage his own client? There are a couple of different ways to think about it – see which makes most sense to you.
One plausible theory is that Tulkinghorn just full-out hates Lady Dedlock and, to a lesser extent, Sir Dedlock. For one thing, we get a couple of clues that he detests women in general. Check out his advice to never get married, and his thoughts that "women were created to give trouble the whole earth over" (42.26). Whoa, dude, settle down.
For another thing, he is in a strange position relative to the Dedlock family. On the one hand, he is a very powerful, incredibly wealthy man, whose political and economic standing probably rivals that of Sir Dedlock. On the other hand, he works for Dedlock and is treated like a favorite servant rather than an equal. He has his own little room to work in, and he comes and goes delivering messages and information to his employer. He is invited to social occasions but spends them hovering in the corner, never really a part of what is going on. It doesn't feel like a stretch to say that he has a lot of disdain for and a generally condescending attitude toward the Dedlocks, and some resentment that the world doesn't treat him with the same kind of awe that it gives to the highborn Baronet. It says everything about how high society feels about him that at his fancy-to-the-max funeral there is no one actually mourning his death, but every noble family sends an empty carriage as a gesture of respect.
This theory relies on analyzing Tulkinghorn's character and personality, and trying to determine what kind of person he is – as if he were a real person. However...
...we could also think about what Tulkinghorn's role is in the novel. Why include such a character? What would have been different if he were a talker? Or if he genuinely cared about Sir Dedlock? Or any number of other different choices Dickens could have made.
One of the many Big Ideas floating around this novel is how secrets are covered up and how they are exposed – in other words, how does detection happen. The invention of the police detective Bucket raises many questions about the morality of finding stuff out about people that they don't want known.
To help us work out those questions, we get many, many different kinds of detectives in the novel. Official ones, unofficial ones, skilled ones, unskilled ones, ones who depend on inanimate objects for clues, ones who read people for clues, and so on. There is a sort of continuum created between just observing the facts and carrying out unsolicited and unethical surveillance. Tulkinghorn is one the best detectives in the novel. He gets to the bottom of its major mystery just before becoming the victim of another one himself. But he is cold, cruel, and sadistic, using horrible tactics in his investigation and just generally being evil. As a result, things don't turn out so well for him. Maybe if he had Bucket's emotional intelligence, he would have known to get Hortense a job, thereby saving his own life with a simple act.