Krum became a seeker for the Bulgarian Quidditch team and he was considered a genius by many, and at the same time, he was at school at Durmstrang. In the 1994 Quidditch World Cup, between Bulgaria and Ireland, he caught the snitch, however, Ireland had enough points to win anyway.
When the Triwizard Tournament came up, he, and some of his students, travelled to Hogwarts to enter themeselves. Krum became the Durmstrang champion. He was often viewed with suspiction, due to Durmstrang's unfortunate reputation of teaching the Dark Arts.
Viktor had a crush on Hermione Granger, continually coming to the library to meet her, but never had the courage to act on his feelings. They eventually went to the Yule Ball together. Ron was furious that Hermione was "fraternizing with the enemy." Karkaroff, who was the headmaster of Durmstrang and didn't admit Muggle-borns, wasn't particulary pleased with this either.
Krum said he "had never felt this way about any other girl." In the second task, Hermione was the thing that Krum had to retrieve.
Rita Skeeter, looking for revenge on Hermione, made up a story for Witch Weekly, which made fun of Hermione by saying she was toying with both Krum and Harry's affections. Harry became sick of telling people that Hermione wasn't his girlfriend.
At the end of Harry Potter and the Goblet of Fire he went back to Durmstrang on the ship. He hasn't shown up again in the fifth book, but major characters rarely disappear and never come back in the Harry Potter books. However, in Harry Potter and the Order of the Phoenix, Hermione was seen writing a letter to him.
Krum's first name is a homonym of "victor" meaning winner.