The vigilante Rorschach doesn't appear in HBO's Watchmen, but his presence is inescapable. In the adaptation of Alan Moore's classic comic, dozens of men carry on his legacy by creating their own Rorschach masks in a quest to take down what they see as a corrupt culture. However, given Rorschach's role in the Watchmen comics, it's hard to say if he would agree with the methods or philosophies of these men.
In the comics, Rorschach is introduced as a masked vigilante investigating the death of a fellow superhero, The Comedian. In the world of Watchmen, costumed heroes have been outlawed unless they work with the government. Rorschach operates outside of the law, viewing himself as a soldier for good unburdened by larger political interests. In Rorschach's mind, there is good and there is evil — nothing in between. Rorschach begins investigating The Comedian's death and stumbles across a conspiracy to kill costumed heroes, even retired ones.
After a brief prison stint, he teams up with his former co-heroes Nite Owl and Silk Spectre. They discover that the culprit behind the deaths of costumed heroes is another hero, Ozymandias. Ozymandias has orchestrated a plan to manufacture a massive "alien" attack and kill millions to unite humanity and end the Cold War, saving the world from a more deadly nuclear apocalypse. Rorschach plans to tell the world what Ozymandias has done when Doctor Manhattan, the only superhero with actual superpowers, kills him to prevent the truth getting out.
However, in the world of the Watchmen HBO series, it appears the truth managed to get out after all.
The final page of the Watchmen comics shows a journal that Rorschach had been keeping about the conspiracy turning up at a right-wing newspaper. It's likely that the publication of the journal led to the rise of the Rorschach-inspired terrorist group seen in the Watchmen trailers, referred to as the Seventh Cavalry, who each make and wear their own Rorschach masks. The Cavalry has taken to attacking police officers, transforming the world from a place where masked crime-fighters are outlawed to a world where crime-fighters are encouraged to wear masks to protect their identities. So while the original Rorschach is dead, his strict definitions of good-and-evil live on the Seventh Cavalry.
However, it's hard to say whether or not Rorschach would agree with the tactics the Cavalry are taking. Rorschach was never hesitant to break a criminal's finger for information or pour hot oil on a prisoner in self-defense, but he rarely turned his violence on authority. As much as he criticized authority and believed that the "the accumulated filth of all their sex and murder" would drown humanity, he never attacked innocent people.
Then again, Rorschach's conservative ideologies line up pretty succinctly with the white men that make up the Seventh Cavalry on the HBO series. We'll never know if Rorschach would've been leading the charge of the Seventh Cavalry or judging them for bringing harm upon others — but the fact is that although he's long dead, Rorschach casts a massive shadow over the Watchmen TV series.