HATERS GONNA HATE
There's no denying Asuka's popularity among fans of Evangelion. Along with Rei, the two girls are plastered over merchandise ranging from figures to body pillows. In a 2010 poll by Newtype, Asuka was voted as the third most popular girl of anime from the 1990's (with Rei as number one), so clearly there's a lot of people who love Asuka. However, it would seem that Asuka's the type of character that one loves or hates: there's quite a few people who really don't like her.Personally, I believe a lot of dislike for Asuka comes from people who have only watched the dubbed version of Neon Genesis Evangelion. Tiffany Grant's performance gives Asuka a very aggressive personality that comes off as being bitchy often. While Miyamura Yuuko's version of Asuka can certainly be bitchy at times, her softer, more feminine voices takes the edge off her attitude and her Asuka tends to be more bossy than rude and obnoxious.
Of course, regardless of who is voicing her, Asuka definitely is headstrong, stubborn and quick to avoid taking responsibility for her own failures and mistakes. For some people, these traits are enough to put them off the character despite the fact that these emotions come from Asuka having experienced fairly severe trauma early on in childhood. They tend to see her as nothing more than an obnoxious brat.
Others seem to hate Asuka purely for the reason that she acts as an obstacle to Rei and Shinji becoming a couple. Although there's no evidence of Shinji and Rei ever having romantic feelings for each other, some fans really like the idea of the pairing. Some just don't like how Asuka treats either character. Even as a fan of her character, I feel that Asuka goes too far in her antagonizing of Rei and Shinji at the end of the series... but considering what she's going through at that point (a severe mental breakdown), I feel more sympathetic toward her than spiteful.
On a similar note, those that hate Asuka tend to really dislike her for her overly sexualized crush on Kaji and her childish hatred toward Misato when she reignites her relationship with him. I'm not entirely a huge fan of Asuka's feelings for Kaji... or to be more specific, I don't really like the way she forces herself on him, so I can understand why people dislike her for it. However, thinking back to my own middle school days, I remember many girls had crushes on older men so I understand where Asuka is coming from. Her anger toward Misato is typical for a young teenage girl in this situation.
While I won't go into a long explanation of why I feel the hate for Asuka is undeserved - the rest of this site does that for me! - I will say that people need to remember that Asuka is a 14 year old girl going through puberty. I think we all can remember this being a time of chaotic unrest: imagine what it would be like for Asuka! She's already suffered so much in the early years of her life, earning her a deep mistrust of other people. The only way she knows how to cope is to seek attention from the others around her and when the attention shifts to someone else (in her case, Shinji), she doesn't know what to do. She has no parental figure to guide her and when she begins to break down, she has no one to turn to for help. It's important not to forget that Asuka is more than what she appears to be on the surface, no different than any of the other characters in Evangelion.