Supernatural, superhero, revenge, action. I wouldn't call it horror, myself.
I bet you might if you read some of the novels. Even in the graphic novels there are some horror elements (the movie is based on a graphic novel and the TV series was based on the movie, though the Skull Cowboy appears in several episodes). The novels... yikes.
The idea is that Eric Draven isn't the only person who ever came back from the dead a year after he (or she) died, to take revenge on the killer(s) of someone - or more (in one novel the Crow is a kindergarten teacher who comes back to exact revenge against those who killed her students in a bomb attack).
As for superhero... Eric Draven isn't like a regular superhero. For one thing, he doesn't get his abilities until after he's dead. And the difference between movie Eric and TV show Eric is that in the movie, the "Crow" look is done with makeup. Eric just becomes much stronger and can't be killed unless his crow avatar is harmed or killed.
Contrast that with the TV show, when Eric physically turns into the Crow (gains the 'look' from the movie). He seems to have the ability to appear and disappear at will, though we never see him do it. A crow shows up, and then Eric turns up out of nowhere (usually scaring the crap out of the cop, Albrecht, who sorta knows what Eric is, yet refuses to treat him as anything but another annoying informant).
TV-Eric Draven does help strangers, sometimes going far out of his way for that. The premise of the TV show isn't "kill the killers". It's redemption for the soul. He does achieve this with one of them... who is murdered anyway. One by one during the single season this show was on, the gang members who did kill Eric and his girlfriend were being killed in various ways. If the show'd had the second season it was supposed to have had, I'd bet the rest of the gang would have died as well.
So yeah, you could consider the TV version of The Crow to be a superhero type of show.
I wrote a LOT of fanfiction for this show, back in the late '90s/early '00s. There were only 22 episodes, but they had a profound effect on my writing and writing habits at the time. I've still got story ideas that have been rattling around, over the past 20+ years.