How is Frankenstein a gothic novel?
This is obviously a big topic, but here are a few pointers to get you started.
1. One characteristic of gothic novels is that they have multiple levels of narration. They do this by using a framing device. We see this in Frankenstein when the story of Frankenstein and his creation is "framed" by the story of Walton. So, we hear Frankenstein's story told to Walton who is telling us the story. At times, the narration is even further removed, such as when the Monster tells Frankenstein his story.
2. Another aspect of gothic literature are the lonely, frightening settings. Gothic literature is "on the margin" of both human psychology, but also geography. The Arctic of course was the literal limit of man's geographical knowledge at the time, but Shelley uses these isolated settings to highlight the loneliness and alienation of the monster, who says:
The desert mountains and dreary glaciers are my refuge. I have wandered here many days; the caves of ice, which I only do not fear, are a dwelling to me, and the only one which man does not grudge.
This shows how he has been rejected by man and labelled as a "monster" - a term which is ironic because it is man through his actions towards the monster that is shown to be acting like a monster.
3. A third aspect of gothic literature is the search for illicit truth and magical mysteries. Frankenstein of course engages in a quest to push the realms of science to their limits and even to go beyond that by his creation of the monster. He sacrifices a "normal" life in his quest and even though he has a wonderful, loving family, goes through periods where he is so focused on his research that he has no contact with them whatsoever. His use of dead bodies adds a gothic feel to his research, and quotes such as: "I entered the diligence into the search of the philosopher's stone and the elixir of life…" make it very clear that he is trying to play God - something that the subtitle of A Modern Prometheus underscores.