Byrne, Ordway and Beatty begin the process of introducing Brainiac into the post-Crisis Superman in Adventures of Superman 438 (March 1988), as Clark Kent and Cat Grant head to the circus, with Adam and Jimmy Olsen as chaperones.
They meet a drunken circus mentalist, Milton Fine, and his naggy wife Janet. Fine calls himself the Amazing Brainiac when he performs, and we do get it confirmed that he truly has some degree of psychic ability.
He tells Adam and Jimmy that his mental powers come from a far away planet, Colu, and a man, Vril Dox, who was disintegrated for disobeying the orders of those in charge. But Vril Dox’s mental essence survived, and has linked with Fine.
His powers come to the fore when he is in Superman’s presence, and Vril Dox takes complete control of Fine, demonstrating abilities far beyond those of a circus sideshow, able to cause great pain to Superman. Superman refers to him as Brainiac, and Dox likes that name, deciding to use it himself.
But as far as Janet is concerned, her husband has simply gone off the deep end, and she throws a bottle at him, disrupting his concentration, and giving Superman the opportunity to knock Fine out.
The man winds up in an asylum, as no one believes his story about Vril Dox, whose control was broken when he got knocked out. But Brainiac will return.
An excellent way to re-introduce the character gradually.