一年の死にたい発言→ one guy has kept on saying that he wants to die.
彼女の図書カード見たら (I think this sentence means the same as 彼女の図書カードを見た時 in his case) →when he looks at her library member's card. (Maybe he is a librarian. Or there are some memories at the library between he and her.)
ダザイとか、な、なんか生まれてきてすみません的な儚げなムードかも出せば(I think it's かもし出せば)→ Showing your fragile feelings like you are kind of sorry to have been born, like Dazai (Osamu*) said in his novel.
*Dazai Osamu is one of the great Japanese authors. His novel called '二十世紀旗手(English title: A standard-bearer of the twentieth century)' begins with '生まれてきてすみません。'
I would say 二十世紀旗手 is not his most popular novel. Actually I didn't know it before I look it up this time.
I mean Dazai is super well known, but I have read only a few of his novels.