Simply put, Hate Fic is Revenge Fic on a series-wide scale. The author sets out to deliberately attack an entire show, either through simply mocking it and its characters or by the techniques of Deconstruction.
When done poorly, this often takes the form of a badly-done Ultimate Showdown of Ultimate Destiny, which attempts to discredit a show based on unfavorable comparison with another rather than by any serious examination of the target series's actual flaws.
Invariably, this leads to cries of "If you hate the show so much, why bother writing about it?"
When this is done in commercially-published work, it's an extreme example of an External Retcon.
Note: It is a Hate Fic only if it covers an entire series. Examples that focus on a single character belong under Revenge Fic.
Anime and Manga
Lala's Words is a To Love-Ru one-shot that decontructs the Unwanted Harem in a variety of ways, such as showing what would really happen if Lala finds out about the rest of the girls loving Rito, how and why they love him and Lala realizing just how uninteresting Rito is. The author outright admits he's not a fan of the show, which could mean he was just playing nice to not gain controversy on hating the anime too much, although that seems to be subtly shown.
These days, it's generally met with either admiration or derisive laughter.
There are even a few fanfics written that are hate fics against hate fics. An example includes Kyrie Eleison, which from the first paragraph states it was written to deconstruct fics like Neon Exodus Evangelion.
The ones for Sailor Moon are the stuff of legend, especially Nav's work. Highlights include one where the Senshi have taken over the world with mind control and are all (about 5 dozen of them) killed by Samus Aran, most without even being mentioned by name, and repeated recurrences of Tuxedo Mask being the only one to put up a fight or the Sailors just giving up before the battle even starts because he's not there — despite these almost universally being based on the anime, where he doesn't even get any attacks, and can just throw roses and give speeches.
In the aftermath of DC Comics' Crisis Crossover, Final Crisis, the internet saw a great torrent of hate fics for it, not to mention the run on Batman that Grant Morrison wrote. The most common criticisms for this is "I do not understand it, therefore I hate and mock it" which is actually a minor theme in the story. People don't seem to realize that this wasn't done by Grant "Justice League and New X-Men" Morrison, but rather was written by Grant "Animal Man and Doom Patrol (or read this multiple times to fully appreciate/understand it), telling a chapter in an epic story that acknowledges all that was written before" Morrison.
Three started as an open Hate Fic of Frank Miller's 300. Being aware of the existence of the Helots and the abuse they got from their Spartan masters (even by Ancient standards), writer Kieron Gillen was incensed by the glorified portrayal of the Spartan warriors in 300 while the Helots weren't even mentioned, and decided to do a comic series where the image of Sparta would be as over the top negative as the one Persia got in 300. However, while doing more research, Gillen realized that he actually found real history more interesting than this approach, and decided to go for a more neutral view, picturing the Spartan characters as flawed, three dimensional human beings, who were almost as much victims of their nation's militaristic culture and expectations as the Helots were of their slave condition.
The Doorstop is one directed at another fanfic, The Open Door, which, if you've heard of it, likely know it's a bit...divisive for readers. Iolaus, the author, read it, and decided he could tell the story better - so he did so, flipping the perspective away from New Chaos and to the more heroic factions, as well as numerous other changes. The result is a bit more even-handed than other examples on this page, although some reviews would claim otherwise.
Right Behind and its sequel Supergeddon for Left Behind.
This fanfic(?) shows how the Federation could go from the TOS to TNG in a disturbingly realistic way.
A fanfiction of the video game Clock Tower 2: The Struggle Within had all the characters slaughtered by the characters from Clock Tower 1. The Struggle Within is probably the least popular of the Clock Tower games.