On the face of it, Linton doesn't seem to have much in common with his old man. He's weak, irritable, effeminate, and with his pallid complexion looks more like his Uncle Edgar than Heathcliff. It's certainly hard to imagine that such an unimpressive specimen could ever have been the fruit of Heathcliff's over-active loins.
Nevertheless, Linton is Heathcliff's son whether he likes it or not (and he clearly doesn't). Having forcibly taken him to be raised at Wuthering Heights, he proceeds to treat him abominably, making it clear to this "puling chicken" that he's singularly unfit to be his son. For most of us, it's difficult to imagine what goes through the mind of someone who'd treat their own flesh and blood this way. But what seems to motivate Heathcliff's cruelty is a kind of jealousy towards Linton for having the kind of secure, mollycoddled upbringing that he never had. And so Heathcliff intends to give Linton a taste of the harshness that he himself endured during his own abusive childhood.
The fact that there's virtually nothing of Heathcliff in Linton illustrates one of the book's themes: the importance of environmental factors in raising children. Heathcliff had a short, nasty, and throughly brutish childhood, and look how he turned out. Whereas Linton grew up in a sheltered, overprotected environment which has created the kind of delicate boy that Heathcliff dismisses contemptuously as no son of his.