Lord Rhoop was one of the Seven lost Lords and a Telmarine lord who opposed King Miraz. These lords did not fear the oceans or forests, unlike most Telmarines, and were of the few who respected the legends of Old Narnia, before the invasion from Telmar.
Rhoop and six other lords were sent to " explore the seas beyond the Lone Islands" by Miraz, in an attempt to get rid of them.
Somewhere along the voyage, Rhoop heard of an Island where Dreams Come True. Somehow or other he ventured to that island, which was covered with a blackness. In that island (called simply the Dark Island), he became ragged, as he discovered that the dreams that come true on the island were unpleasant, terrifying nightmares.
Finally on the island of Ramandu, he was given a dreamless sleep by Ramandu, and slept with his former comrades, Argoz, Revilian, and Mavramorn. At the end of The Dawn Treader, it states that when Caspian came back to Ramandu's island, the lords had awoken from their sleep.
In the film, Rhoop was more or less the same as he was in the book, with some minor differences.
He and his fellow Lords had fled to the Lone Islands, to escape from Miraz. While they were there, they learned of a Green Mist that was abducting people, and each swore they would find a way to stop it. They were told by the magician Coriakin that the only way to destroy the Mist was to lay their Seven Swords at Aslan's Table. They each travelled long and far, losing some of their number along the way, until finally they reached Ramandu's Island, where Aslan's Table was.
Unfortunately, by the time they arrived, they only had four of the swords. Three of them went half mad, threatening violence upon each other, which was forbidden at Aslan's Table, so they were put to sleep. Only Rhoop remained, and he ended up going to Dark Island, possibly because that was where the source of the Mist's power lay, and he sought to go fight it himself. Alas, he severely underestimated the Mist's magic, and it drove him mad for years, before he was finally rescued by Caspian and the crew of the Dawn Treader.