Was Rorschach suicidal?

The movie, due to time limitations, does not show all the conversations between Rorschach and the psychiatrist. Rorschach dislikes him because he keeps calling him Walter Kovaks. The doctor tries to understand why Kovaks created Rorschach, thinking of it as just a "dissociative identity disorder", while Rorschach sees in a different way: until that moment (when he realized the girl was dead), he was "Kovaks pretending to be Rorschach". From that moment, only Rorschach existed.

So he is just explaining that he does not relate to his old name anymore, he's Rorschach all the time.

As @Paulie_D says, he's not suicidal, he is just in a dead end: He will never lie about the responsible of the killings even if that causes a Word War III. And he also knows that Dr. Manhattan has already made up his mind and he won't let Rorschach tell the truth, so he prefers to die rather than break his principles.

At the end he begs Dr. Manhatten to kill him.

Does he?

Rorschach and Nite Owl are two sides of the same character root inspiration - Batman (not discussing the Charleton characters here)

Nite Owl is the rich, 60s toy/gadget part of the Dark Knight story, Rorschach is the early/late revenge driven version.

Rorschach is much more successful as a vigilante than Nite Owl as he has the strategic mind required, and Nite Owl acknowledges this:

He also won't let a small thing like the end of the world stop the truth coming out:

Before Rorschach and Nite Owl head to Antarctica, he posts his journal to the "New Frontiersman" saying whatever happens from that point is down to Veidt, and is a plot.

Once Veidt's plan is enacted Rorschach knows the others will remain silent, and will stop him from speaking in the name of peace.

He therefore decides to sacrifice himself, to distract the others from stopping his journal reaching the public (which is about to happen at the end of the story). His last message to the New Frontiersman is he expects to die.