What is "Gothic" about the weather at the beginning of Dracula?
In literature, "Gothic" means stories about the supernatural, or that privilege irrationality and emotion over reason. Usually Gothic literature takes the form of horror stories, like Dracula, set in wild, picturesque places. Take a look at this eNotes page for a more detailed discussion of the Gothic genre and its origins.
Part of the "supernatural" element of the Gothic is the idea that the supernatural power at the heart of the story is somehow connected to, or can effect, the weather. For example, as Harker is on his way to meet the Count's carriage at the Borgo Pass, he passes a place called "God's seat," a mountain of tremendous magnificence:
Beyond the green swelling hills of the Mittel Land rose mighty slopes of forest up to the lofty steeps of the Carpathians themselves. Right and left of us they towered, with the afternoon sun falling full upon them and bringing out all the glorious colours of this beautiful range, deep blue and purple in the shadows of the peaks, green and brown where grass and rock mingled, and an endless perspective of jagged rock and pointed crags, till these were themselves lost in the distance, where the snowy peaks rose grandly. Here and there seemed mighty rifts in the mountains, through which, as the sun began to sink, we saw now and again the white gleam of falling water. One of my companions touched my arm as we swept round the base of a hill and opened up the lofty, snow-covered peak of a mountain, which seemed, as we wound on our serpentine way, to be right before us. "Look! Isten szek!"—"God's seat!"—and he crossed himself reverently.
Once they get past "God's seat" (and symbolically enter a land without God), the atmosphere changes:
And at last we saw before us the Pass opening out on the eastern side. There were dark, rolling clouds overhead, and in the air the heavy, oppressive sense of thunder. It seemed as though the mountain range had separated two atmospheres, and that now we had got into the thunderous one.
The weather foreshadows the terrible appearance of Dracula's carriage, and the nightmarish experience Harker is about to have as Dracula's "guest"!