Read other examples of Gothic literature and horror novels of the late 19th century. "The Phantom of the Opera" is a late example of Gothic horror -- emphasizing foreboding settings, doomed romances and deformed figures who struggle with their inner monstrosity. It shares many thematic elements with other horror novels of the period such as "Dracula" and "Dr. Jekyll and Mr. Hyde."
Research Parisian theater of the late 19th century. "The Phantom of the Opera" is set at the famed Paris Opera House, which has a rich and romantic history. In his introduction Leroux claimed that he based the story on actual legends about an "Opera Ghost," and even without such claims, the Opera House plays a vital role in how the story unfolds. You may also wish to research the Grand Guignol theater in Paris, which specialized in gruesome and horrifying performances. The Grand Guignol opened in 1897, and was flourishing when Leroux wrote his book. It influenced the horrific nature of the text.
Check the translation of the particular version you're reading. Leroux originally wrote the text in French; an English translation was first published in 1911, with several additional translations published in the 1990s and early 21st century. Each translation is slightly different, and reading the book in its original French is a different experience than reading a translation. Opinions differ on which translation is best, though a 1996 version edited by Leonard Wolfe bills itself as "the definitive, annotated edition of Leroux's classical novel" according to The Phantom of the Opera.com
Read the novel without the benefit of seeing either the stage play or any of the various film translations. If you've already seen the story in another medium, wait for a time before picking up the book. It will help you look at it with fresh eyes.
Evaluate the way Leroux presents his characters and the storyline as you read. In particular, note the way he emphasizes the tragic nature of the Phantom himself and the heartbreaking denouement that differs significantly from popular perceptions of the novel.
Watch a stage or movie version after you've read the book and compare the difference between them. Consider how subsequent productions -- particularly the Lon Chaney film and the Andrew Lloyd Webber musical -- have influenced perception of the novel and Leroux's original characters.