Mary Shelley's novel Frankenstein is considered both Romantic and Gothic. Given the horror of the novel, the Gothic elements of the novel are relatively easy to identify. This said, the Romantic characteristics of the novel can be a little harder to identify if one is not familiar with typical Romantic conventions.
Romantics glorified nature, dreams and visions, the power of the individual (over that of the whole/society), and intuition (one's emotions and feelings about something). In regards to the character of Victor Frankenstein, Romantic characteristics exude from him.
Dreams and Visions
Victor has two dreams which prove to foreshadow coming events. His first dream comes soon after he finds success in re-animating life. He collapses from exhaustion and falls into a troubled sleep. During this, Victor dreams of embracing Elizabeth. As he leans in to kiss her, Elizabeth transforms into his deceased mother.
In the second dream, Victor is on his way home from Ireland (after being cleared of Clerval's murder). He dreams of the Creature wrapping his hands around his neck in an attempt to strangle and kill him. While the Creature is not directly responsible for Victor's death, he does play a substantial part in leading him (Victor) down the path to his destruction.
The Power of the Individual
Victor proves that he possesses great power. The first time readers see this power is when he surpasses his professors at Ingolstadt. Finding they can no longer teach him anything, Victor goes out on his own to re-animate life. With his success, Victor shows his power.
Victor also proves to be powerful in regards to the numerous deaths and challenges he faces in his life. He loses everyone close to him, including his new wife, and still finds the strength to track the Creature to the ends of the earth.
Lastly, Victor's success at re-animating life proves him to be "god-like." Given that, according to Christian ideology, God is the only one who can create a new species (as Victor refers to the Creature), Victor has now placed himself on the level of a god.
Intuition is ruled by feelings and emotions about something. That said, Victor possesses true intuition. In his studies, Victor was passionate. The deaths of his family members and Clerval, Victor's immense emotion forces him into illnesses. The intensity with which he wanted to find success proves to show exactly how emotional Victor was (given he removed himself from everything around him and even refused to care for his own health).
Perhaps the most Romantic characteristics Victor possesses is his dependency upon nature. It is nature which heals him from all of his illnesses. The coming of spring, and its regenerative nature, proves to Victor that hope can always be found.