Many popular ones will stay true to the original characters' personalities. So, Christine might be shown as kind and a bit conflicted, and Erik as complex and passionate. This helps the readers to feel more connected to the story as it aligns with what they already know about the characters.
One popular trope is the 'Forbidden Love' aspect. Since Erik is often seen as an outcast, their kiss can be a very intense moment within the context of a relationship that society might not approve of.
You can try fanfiction websites like Archive of Our Own (AO3). There are often a large number of fan - created works there. Just search for 'christine and erik kiss' in the search bar, and it will show you relevant stories.
In a different version, Christine gets lost in the opera house and Erik is the one who finds her. He could have been watching her from afar all this time. Instead of scaring her, he gently leads her to a safe place. They start talking about their dreams and fears. Christine realizes that Erik is not the monster everyone makes him out to be. This new understanding blossoms into love as they continue to meet secretly and share their innermost thoughts.
A simple fanfiction could be that Erik writes a beautiful song for Christine. When she hears it, she is deeply moved. She seeks him out to thank him. This encounter starts a series of meetings where they slowly fall in love. Their love is based on mutual respect for each other's talents.
You can try fanfiction platforms like Archive of Our Own (AO3). There are a large number of fanfictions about Erik and Christine's kiss there. You can simply search for 'Erik and Christine kiss' in the search bar, and it will show you various works with different storylines and writing styles.
There's also the theme of sacrifice. Sometimes Erik sacrifices his own desires for Christine's happiness, and vice versa. It adds a layer of depth to their relationship. And the exploration of their different social standings, with Erik being an outcast and Christine being a more 'normal' part of society, is another common theme in these fanfictions.
A unique way to develop their relationship could be through a shared dream or vision. They both see a future together that is full of love and acceptance. This shared experience makes them more determined to be together. Christine starts to see past Erik's physical appearance and focuses on the beauty of his soul. They then start to explore their love in a more physical way, like holding hands and then gradually becoming more affectionate.