I remember this as a paperback novel. Main character had jet black skin and a psychic "danger" sense. It was a post nuclear world. 70s or 80s book (I think). Main character was adult human male. I think he was not actually black by descent, but because he was some sort of mutant. I remember him fighting gangs, sort of a Mad Max knock off.
Any chance that you're thinking of Telempath, by Spider Robinson? Here is a cover scan. (I found there are several to choose from -- several editions over the last 40 years, I guess.)
Here are some items that make me think it might match your description, allowing for some margin of error.
The novel-length version was published in paperback in 1976. (It was an extended version of "By Any Other Name," which had previously won a Hugo for Best Novella.)
The main character (and narrator) is Isham Stone, a young man of African-American heritage. I can't remember if we were told that his skin was "jet black," though.
He did live in a postapocalyptic world, where the human population was only a fraction of what it used to be -- but nuclear war was not to blame.
Instead, due to a "mad scientist" doing something reckless (on purpose!), everyone had suddenly developed an incredibly enhanced sense of smell, and the massive sensory overload had caused a large portion of the human race to go crazy and die soon after being exposed to the catalyst that was spreading through the atmosphere. (In some cases, people died because others went berserk and killed them.)
The protagonist grew up well-adapted to this new set of circumstances, but the result was that instead of having a "psychic sense" per se, he routinely learned all sorts of interesting things (including other people's moods) from his nose; things which you and I either would never detect, or would only detect after we got very close to the source of a particular scent; whereas Isham Stone might smell it from hundreds of meters away. (I forget the exact range and limitations of his sense of smell; it's been a heck of a long time since I read this material.)
Sounds like a series of novels called Traveler by DB Drumm (pseudonym of John Shirley). The book in particular you are thinking of could be To Kill a Shadow which is the 4th book in the series. The main character in these books is a Mad Max type man named Traveler that has a kind of extra-sensory ability thanks to being exposed to neuro-toxins before the apocalypse. The main antagonist in book 4 is a mutant biker with pitch black skin.