Keith Haring was everywhere in the 1980s. All over the walls of the New York subway, all over T-shirts, all over posters, all over dresses and all over Grace Jones’s body. You could wear him on your lapel. He was there when you needed him and there when you didn’t. He was in all the galleries and nightclubs and he invaded your house. He was all over MTV and he wanted to be all over the Berlin Wall, a section of which he painted. Haring was joyous, riotous, funny and angry, this gawky skinny kid with the glasses and a Sharpie pen. And then he was gone, dead at 31 from Aids-related illness in 1990.

Now he fills the top floor of Tate Liverpool and the gallery shop. His little bouncy figures climb the columns in the cafe. With his outlined boing-boing beings and miraculous babies, his barking dogs and his three-eyed faces, his boys in heat and his general malarkeys, he took potshots at God, sexual repression, America’s culture of guns and the dollar, and much besides.

Ignorance = Fear, 1989. Photograph: © Keith Haring Foundation

There was more to Haring than his ubiquitous cartoons, and this travelling exhibition (it goes to Brussels and Essen), mounted with the co-operation of the artist’s estate, unpacks his work from its early beginnings to his outspoken responses to the Aids crisis. There are videos of early student performance works and projected images of the 80s New York club scene. Andy Warhol, William Burroughs, David Hockney, Jean-Michel Basquiat, Tom Verlaine and Robert Mapplethorpe pass through, along with all the denizens of the downtown dark.

Heaving with malevolence … Untitled, 1983. Photograph: Krause, Johansen/© Keith Haring Foundation

And always there were the drawings, from early, art-school abstractions and filmed performances, drawings made in public on the subway (for Haring, drawing was a kind of performance art) and drawings often used as posters, protesting against apartheid, railing against religious intolerance and bigotry, homophobia and racism. He celebrated Coming Out Day and had a chirpy cartoon condom promote safe sex.

Haring made a great deal out of what appears an extremely limited lexicon of forms and a basic, though singular, style. The thick, simplified line, the speed and certainty of his one-shot drawing approach, wouldn’t seem to offer much leeway. But the show is full of variety, for one thing in the way Haring kept complicating and developing his bestiary. He presented his painted drawings in a UV-lit gallery basement, against walls striped red and yellow, with blasting music. He drew stories and parables, cheeky same-sex couplings, almost medieval scenes of torture, torments, monsters, Bosch-like penis-beings, choking cockerels, angels and maggots, flying skulls and tiny humans ripped apart.

A recreation of Haring’s UV-lit gallery basement. Photograph: © Mark McNulty

His later work (in a career lasting barely a decade) teems with life and death, pluggings and unpluggings, gods and demons. What began with wry smiles, as a buoyant, seemingly benign art, ended up heaving with malevolence. Going through the show I felt like Dante being led into the circles of hell by Mickey Mouse. Walking past walls covered in club flyers, seeing footage of Haring handcuffed by a transport cop for drawing on the subway, passing through the Mudd Club and Club 57, then marching on Washington to protest government inaction on Aids. In the 80s, Haring wrote the writing on the wall: Crack is Wack, Ignorance = Fear and Silence = Death. He drew flying saucers training their beams on the city, and a prone body spewing a monstrous cloud of chaos into the sky.

The modern apocalypse is no less filled with demons as any medieval vision of the end. For a gay man in the 80s, the end was all too present and personal. Through all this, Haring’s art – and it was art of a high order when seen in full – never lost its grip. Making work increasingly terrible and apocalyptic in its own visions, he somehow still manages to exhilarate us through it all. As much as Haring was an artist of his time – he seems almost to typify it – the show feels horribly prescient. Same end, different apocalypse.