The relationship between the two half-brothers Elijah and Klaus has frequently changed between being best friends to enemies and back again, due to Klaus constantly daggering his half-siblings. Elijah and Klaus finally begin to repair their brotherly bonds during the third season of The Vampire Diaries. In The Originals, their relationship swings from partners in crime to enemies, usually based on Klaus' mood. Upon returning to New Orleans, Klaus saw his relationship with Elijah as a weakness that could be exploited by his enemies, and often took out his frustrations on Elijah as a result. However, Elijah, unlike Rebekah, is unwilling to give up on Klaus and insists that the only moment he will stop looking for his half-brother's redemption is when Elijah believes there is no chance left of his redemption to be found.
In the first season of The Originals, it was shown that while Klaus cared for his half-brother, he was willing to dagger him in order to regain his title of king of New Orleans, and even went so far as to bite him when Elijah falsely accused him of planning to use his child to sire more hybrids.
In the second season of The Originals, Klaus and Elijah's relationship has improved greatly, as the strife they had dealt with involving the witches (led by Genevieve) and the Guerrera werewolves, in addition to being forced to send away Klaus' daughter Hope, had caused the half-brothers to start trusting each other more and generally treating each other better.
Unfortunately, in the end of the second season, their relationship becomes worse than ever, when Elijah daggered Klaus and allowed Alex to try and take Hope away from him, in retaliation Klaus kills his half-brother's friend Gia, and daggered Elijah with Papa Tunde's Blade, their relationship becomes estranged/broken after this, as they both want nothing to do with each other any more.
By the end of the third season, they reconcile. Early in the fifth season their relationship has become strained again due to Elijah's erased memories and Alex's death, however, at the end of the Til the Day I Die; they reconcile again.