Considering the use of the word "agitated" in this line, the most likely meaning of the word "convulsive" is: "spasmodic."
The line mentioned is from the moment Victor Frankenstein realizes his experiment was starting to work out, since the (previously) inanimate matter displayed some movement by opening its eye and moving its limbs suddenly. The author wanted to give the reader a feeling of being before a dying person in their convulsions and spasms. The most scary effect it has is that that "person" is not about to die; in fact, the "person" is about to come to life.