Saya Takagi from Highschool of the Dead was really whiny. Thinking she was better than everyone.
I haven't watched Kuroshitsuji in ages, but I remember that Elizabeth/Lizzy REALLY got on my nerves... Ugh!
Misa Misa from Desu Noto was kinda whiny as well...
Sakura in Naruto was just more annoying and clingy (much like Lizzy) than she was whiny. I honestly didn't like her at all when I was younger and watched Naruto an awful lot. :P
Latvia in Hetalia was sorta whiny. He cried a lot and always seemed sad, timid, you name it.. He was scared of Russia so I get it.
Yukiteru Amano.. He was whiny.
Ukraine from hetalia