The first Buffy character that comes to mind is Kennedy. Like Joyce said, she acted like she knew everything from the getgo, and every time she spoke, I just wanted to slap her in the face. And I just thought she was a terrible girlfriend for Willow. They were hardly even friends, Willow always looked like she felt terribly awkward when they were talking, they were nothing alike, they had NO chemistry, Kennedy seemed so much younger than her, their relationship progressed far too quickly...I could go on. I know, a lot of people say, "Oh, you just don't like Kennedy because she's not Tara!" Wrong! I don't like her because she's so annoying!
On that note, I was incredibly annoyed by almost all of the Potentials. I thought season 7 was really good until they came along and completely ruined it. Rona really grated me sometimes, and even the Dania Ramirez character got on my nerves (although she gets extra points for being on Heroes now, even if Maya isn't really a great character.) I did, however, like Vi and the Chinese girl who didn't speak any English. Neither of them got enough screentime. And Amanda was alright at times, but I think she was just good compared to the other Potentials...if she had been the only one, I think we'd all hate her.
I also didn't like Warren, but to be honest, that's just because he killed Tara (although I did think the episode where he killed Katrina was incredibly boring.)
On Angel, I don't like Connor or Holtz. Do I even have to explain why I didn't like Connor? He whined constantly, he always got himself involved in things that were none of his business, he acted like he was "all that," and he slept with Cordy, who was practically his mother! Gross! I have to say, however, that he really redeemed himself in season 5. He was great in the two episodes he was in, and from what I've heard, he's good in After the Fall (I still need to get ATF, especially because I love Buffy season 8.) Holtz just kind of took up screentime, and, though he was well-acted, he was just boring in general. His actions were justified in the beginning, but after a few episodes, it was just, "We get it! You don't like Angel! Now get out!" I think I'm the only person that didn't hate Eve, but I certainly didn't like her. No one can replace Lilah.
Like Joyce said, all of the characters did a few things that annoyed me, particularly Xander, Dawn, and Buffy. I was always incredibly annoyed by Xander's reactions to things that didn't go his way, since he always just freaked out. Dawn complained a bit too much for my taste in seasons 5 and 6, although her complaints were definitely justified (and she was great in season 7.). Above all, though, I just hated Buffy's vanity. I understand, her superiority complex was a major part of season 7, but she had it since the show first started! As Willow would say, at times, she deserved to have "every square inch of her ass kicked."
Also, what's with all the Joyce hate?! She was a great character! Not only was she necessary, but she was a very realistic portrayal of a single mother put into those situations.