The Bell Jar by Sylvia Plath (who killed herself--the author did). I didn't read this. All her material *may* contain suicidal stuff because she is known for this because she was depressive and did kill herself. So check her out as an author.
The Catcher the the Rye by J.D. Salinger. Read this in high school. I did not like it or relate at all despite that it was other students' favorite book, including my then boyfriend, who loved it.
Suicide Notes by Michael Thomas Ford. Did not read this either.
Edit: From my book list I read, I *think* one or all of theses books from Chuck Palahniuk *may* have a suicidal character. He is the author of Fight Club, and in that book, even, the character is suicidal, but if you saw the movie, it is in a totally different way. So the other books from him to maybe look at are: Invisible Monsters, Choke, and Survivor. I read all these back in '02-'03, so my memory is shoddy. He is a very particularly styled author in his voice, so you may or may not like him. He's not sad but is what I might say is sort of sarcastically negative in an bit of an immaturish, comical way. I think he appeals mostly to younger men in their 20s, or did at the time.
Darn, there are more books on my list: House of Sand and Fog by Andre Dubus: this is a very good book and quite depressing. He is a very good author. There is also a movie, and that is good as well. I do recommend this.
My husband has met this author IRL but never read this book of hers: The Day I Killed James by Catherine Ryan Hyde.
Life as a House is a movie I saw that I think is also a book. It was a good movie, so I think it would be a good book.
Joe Hill's a Heart Shaped Box is not sad--it is horror--but the monster in it tries to get people to kill themselves. It is a good book by a good author--Stephen King's son.
I am sure I can think of others over time I have heard of, but I do not think I can recall any more I actually read. I have a book list going back 20 years of books from my 20s on I have read and don't see others I recollect with suicidal things, but will keep thinking for you because I only record books I READ cover to cover, and I put A LOT books down early on when I don't like them.