This is actually a really interesting question! The format of the story is quite multimedia in nature. Leroux narrates many of the events as he imagines them happening (he gets quite deep inside Raoul's head, for one thing), but in-text he also incorporates transcripts of court proceedings, letters (e.g. the notes to Moncharmin and Richard from the Phantom), and accounts entrusted to him by others, most prominently near the end with the Daroga's side of the story. At one point, there's even a rather strange switch to a script format, during which Philippe shows Raoul an article in L'Epoque (a newspaper) about the budding romance between Raoul and Christine. Leroux also makes a reference at the end to Christine's papers, presumably her account of the story.
Regarding genre: though the story was published in serial form from 1909 to 1910, the novel itself would appear to take place sometime between 1870 and 1885, according to the letter Mme. Giry reads to Moncharmin and Richard. It dovetails with the Gothic genre in many ways--most obviously with a malevolent supernatural presence, a virtuous and innocent yet ultimately powerless heroine in Christine Daae, the Phantom's obsession with Christine's innocence, and the attempt at the end to redeem the Phantom through the power of Christine's innocence and kindness.