Out of the many villains who populate the DC universe it's Martian Manhunter who is his true archnemesis. That's because the Manhunter's backstory is directly linked to Darkseid in more than a few surprising ways.
Despite being one of DC's oldest superheroes, the Manhunter from Mars rarely takes the spotlight in comics. In fact, when DC rebooted their universe for the Mars was actually connected to the Anti-Life Equation and that Darkseid himself may have been responsible for the destruction of Mars.
One of the first changes Ostrander and Mandrake made to Martian Manhunter's backstory was the introduction of his evil twin Ma'alefa'ak (also known as Malefic, because comic books are rarely subtle). Malefic was responsible for unleashing a plague of fire, the Martian weakness, to slaughter his own people until he and J'onn were the only Martians left. It's revealed during the run that Malefic did this because he had begun worshipping Darkseid and the New Gods of Apokolips But that's not the only connection between Martain Manhunter and Darkseid-a three part flashback issue from Martian Manhunter #33-35 shows how, prior to its destruction, Darkseid's minions attempted to make with Martian civilization as part of a scheme to obtain the Anti-Life Equation.
When the forces of Apokolips came to Mars they came as ambassadors and pretended to befriend the Martians, including J'onn. But their true intentions were to betray the Martians and steal their philosophy and science. Martians in DC worship the concept of life itself and believe in a universal goodness they call the life equation. Based on this fact, Darkseid concluded that the life equation must have an opposite, and so it was the Martians who accidentally confirmed for Darkseid that the Anti-Life Equation was real. It's also during this time that J'onn meets Kanto, Darkseid's assassin who becomes a personal enemy of the Martian Manhunter. Eventually, however, Darkseid's forces reveal their true colors and launch an invasion of Mars, kidnapping countless Martian children to be indoctrinated into Darkseid's armies. And so J'onn swears he will go to Apokolips to free the children and stop Darkseid no matter the cost.
Martian Manhunter does succeed in freeing the kidnapped children from Apokolips, but he is unable to stop his brother from destroying the Martian race. It was never directly stated that Malefic killed the Martians on Darkseid's orders but the implication is that he thought he was following Darkseid's will. Not only does J'onn have a deeply personal reason for considering the lord of Apokolips his arch-enemy but there is another thematic reason why Martian Manhunter is the perfect hero to contrast Darkseid. Mars prior to its destruction was a world full of life, a world where Martians deeply valued love and trust and bonded with one another on a mental and spiritual level. In order words, it is the exact opposite of Apokolips, a world ruled by fear and discrimination where people are taught to hate one another and worship tyranny. As the last surviving Martian the Martian Manhunter considers himself to be the inheritor of Mars' philosophy of love, making him the natural rival to Darkseid and his evil.
