The rhino is made of clouds, so it probably can't be harmed.
The rhino seems to have the ability to control the weather because when it is seen, a strong wind comes and a storm cloud appears; and when it arrives, lightning appears afterwards followed by thunder.
The rhino also uses fear as a weapon; the more a person fears it, the stronger it gets. In fact, it represents James' fear.
In the film, the rhino came out of nowhere and gobbled up James' parents, forcing James to live with Aunt Sponge and Aunt Spiker.
Spiker and Sponge used the rhino to scare James. It is unknown whether it was once a real rhinoceros when it killed James' parents or if it is just a spirit haunting James (more likely the second one since rhinoceroses are herbivores and do not eat meat).
When James and his insect friends finally reach New York, the rhino appears and tries to slow down their progress. Slowly, it approaches the giant peach, using the power of fear. When James manages to muster up the courage to say he isn't afraid of the rhino, it ultimately dissolved, but not before shooting a final bolt of lightning at the giant peach.