During the Triwizard Tournament of 1994, Krum was picked by the Goblet of Fire to be the Champion of Drumstrang. During the first task, he got the Golden Egg from his foe, the Chinese Fireball. He exhibited good use of the Conficiatus Curse, striking at the dragon's eyes, it's weak point. However, in the process, the dragon trampled on most of the other eggs, lowering his marks from most of the judges. Karkaroff, his head master, still awarded him highly.
It was during this time that Krum asked Hermione Granger to the Yule Ball. Hermione had noticed Krum in the library before, although that was mainly because of the gaggle of giggling girls that followed him around, making her studying much harder. However, it turned out that during Krum's visits to the library, he had been plucking up the courage to ask her out.
In the second task, Krum had to swim to the bottom of Hogwarts' lake and retrieve that which was most precious to him, Hermione. He showed good work in transfiuring his head to that of a shark's. He recieved help from Harry Potter in the form of a sharp rock (in his current state, he tried to bite the ropes off of Hermione). He returned successfully, although he did it outside of the given time limit.
By the time of the third task, Viktor had 80 points, five points behind Cedric Diggory and Harry Potter who were tied for first. This meant Krum would enter the maze third, after a simultaneous entry from the two Hogwarts' champions. They would be followed by the Beauxbatons champion, Fleur Delacour.
We also meet, briefly, Viktor Krum's parents in the chapter "The Third Task" in Harry Potter and the Goblet of Fire. He seems to have inherited his mother's dark hair, and his father's large nose.
At the end of Harry Potter and the Goblet of Fire, Krum returns to Durmstrang on their ship which rises out of the lake beside Hogwarts. He does not make appearance in the fifth book, but Hermione writes at least one long letter to him that year, showing that they maintain contact. Contrary to his invitation, however, Hermione does not visit him in Bulgaria.
Krum briefly returns in the 7th book Harry Potter and the Deathly Hallows, as a wedding guest of Bill Weasley and Fleur Delacour. He talks to a disguised Harry about the lack of available women before having a heated discussion with Xenophilius Lovegood. He recognises a symbol that Xenophilius wears around his neck as the mark of Grindelwald, which is engraved into a wall at Durmstrang; the symbol is later discovered to actually be the mark of the Deathly Hallows. Krum doesn't seem to mind when Ron asks Hermione to dance at the wedding, and comments, to Harry's annoyance, on Ginny's attractiveness.