In Vampire Diaries S02E21 and also a few near-by episodes, it is said that Klaus is cursed - as he is both a Vampire and a Werewolf and more powerful than both.
How is it a curse and not a blessing?
From Vampire Diaries Wiki:
Hybrids are obligated to serve Klaus because his blood created them and they feel loyal to him because he freed them from the pain of turning every full moon. They have limited free will, along with lack of control.
This lack of control and their susceptibility to mind-control is a curse which only hybrids have.
EDIT: In light of OP's edit, narrowing down the discussion to Klaus only, the possible explanation is the worst of both the species - Vampires and Werewolves. The thirst for blood, as well as the pain of turning into a wolf.
It's a curse because he has the disadvantages of both. He thirsts for blood while suffering the effects of turning into a Wherewolf.
It's called a curse because the original witch (his mother) did a spell on him to suppress his werewolf part. At the end of season 2, he broke the curse. He could turn freely thereafter and the the curse no longer exists.
All the ups are pretty snazzy but the downs are pretty bad as well. Also you would have to serve KLAUS, which isn't the best thing to be doing considering how many people wanna kill him. If you can break the sire bond you still need to control your thirst for blood. Let's just remember back to what Tyler had to do to break the sire bond. Lots of pain involved. Unless you love pain and Klaus yep its a curse.