Zenigata has made it his mission in life to arrest Lupin. He would love to arrest the other members of Lupin's gang, but Zenigata is really after the mastermind himself. Lupin and Zenigata appear to be the worst of enemies, but they are, in a manner of speaking, friends; something Lupin shows openly (often by greeting Zenigata with mock affection), but it is an idea Zenigata is extremely reluctant to entertain. Similarly, he is often awed by Lupin's genius, but refuses to show it.
In the movie Mystery of Mamo, it is mentioned that Zenigata has a daughter named Toshiko, but this isn't developed any further. He's unable to settle down because of his eternal pursuit, and although he sometimes longs for female company he has no luck with women - in no small part due to his inability to commit himself to anything but chasing Lupin. The Manga series reveals that he has a grandson, a student at Todai University, who fiercely opposes Lupin's attendance, though by the end of the arc, he has largely given up on having Lupin expelled outright.
The anime Zenigata is a quite sensitive person and often cries in emotional situations, particularly when he thinks Lupin has done something out of consideration for him, or when finding out Lupin is OK after worrying about him. On occasions when Lupin appears to be dead, he mourns him more expressively than anyone else; partly from genuine grief, partly because he no longer has a goal in life. Because of this singular ambition, he is never particularly displeased when Lupin manages to escape his custody, since that means the hunt will go on. In one instance in the manga, Zenigata even goes so far as to cryogenically freeze himself so that he can continue to chase Lupin in the future, though both his and Lupin's cryopods malfunction and they are stuck in the present.