The Romantic period is basically a rebellion against the ideas of the rigid and stuffy 18th Century. Therefore, some qualities that signify Romantic works are: love of nature, importance of the individual (rather than society as a whole), imagination, the supernatural, and an emphasis on emotion (rather than logic/reason).
In the letters, Walton repeatedly describes the beauty of his surroundings. He also mentions his goals--which are all about the success of his individual wants and glory. He is also more emotional than most men we know. He is woeful that he is among people with whom he can not be friends, that they can not possibly understand him and his pursuit of knowledge.
The rest of the Romantic factors will come out throughout the rest of the novel. Enjoy it...it's one of my favorites! :)