Richard I, king of Britain, wanted to wrest control of Jerusalem from its conqueror, Saladin. To attempt that objective, he joined the Third Crusade.
In this video clip, from the BBC's recreation of "Richard the Lionheart," the new King embarks on his trip to "the Holy Land."
The idea of British royalty, assuming a leadership role in the Crusades, did not originate with King Henry II (or his sons). There is an interesting back story which began in Jerusalem. The account is split into paragraphs, here, for easier reading:
At the end of January 1185 Heraclius, the patriarch of Jerusalem, came to England to implore Henry's aid for the perishing realm of Palestine. King Baldwin IV was dying; after him there was but one male heir left of the blood of King Fulk of Anjou and Queen Melisenda, and that one was a little child.
From the story as told by Gerald it seems plain that Heraclius aimed at something more than merely persuading Henry [the II, father of Richard and John] to take the command of a crusade; his project was nothing less than a transfer of the succession from the younger to the elder Angevin line - from the infant son of Fulk's graddaughter to a son of Fulk's grandson, Henry.
When the king of England, after taking counsel with his "faithful men," declared that he could not in person undertake the deliverance of the Holy Land from its enemies, Heraclius still persisted in his other request; he implored Henry to send at least one of his sons - if even it were only John - "that from this scion of the Angevin house the seed royal might be raised up and spring into new life."
The king, however, would not listen. John, it is said, was inclined to embrace the patriarch's suggestion, and threw himself at his father's feet to beg his consent, but in vain. (John Lackland, by Kate Norgate, pages 11 and 12.)
Instead ... when Richard (the third son) became King, he took it upon himself to personally participate in the Third Crusade. His own lands were managed by regents - including his mother, Eleanor of Aquitaine - while he was away.
Clip of "Richard the Lionheart," from the BBC's series "Heroes and Villains." Online, courtesy BBC's Channel at YouTube. Copyright, BBC, all rights reserved. Clip provided here as fair use for educational purposes and to acquaint new viewers with the program.
Quoted passage from John Lackland, by Kate Norgate.