Victor Frankenstein begins the story (which is actually near the end of the story, chronologically) in the Arctic because he has chased the "daemon" to whom he gave life there. One day, when their ship was hemmed in by ice, Captain Walton and his men saw a dog sled in the distance, going north, and driven by something that looked like a man except that it was very much larger.
The next day, when the ice has begun to break up, a man who we later find out is Victor has floated up to the ship on a piece of ice. Although he only has one living dog remaining to pull his sled, and despite the fact that "His limbs were nearly frozen, and his body dreadfully emaciated by fatigue and suffering," he will not come aboard the ship until he knows which way they sail. Despite the fact that he would certainly die on the ice, alone, he will only get on the boat if it is going in the same direction as the creature, who he intends to pursue until one or both of them are dead. Once he learns that they travel "towards the northern pole," he is then willing to come aboard.
This is why he's in the Arctic, to seek revenge on the daemon who killed almost everyone he ever loved.