Personally in his way of thinking, life, and modus operandi, I don't think Dick Grayson is the type of person who would have an arch-enemy. I mean jerk villains have even come at him with these personal grudges and history "from way back" trying to make nemesis-status stick and it never sticks. Dude is an upbeat, believes in the good in people piece of teflon.
From a more meta, authorial point of view ... it's hard to have an arch-enemy sometimes. Batman is very cool. Joker is kind of equally cool in his own way, or "neat" anyway. Great look, good hook. Kind of small-time considering Batman Justice Leagues every day but Joker only Dooms once in a great while. But it's personal. It's a local beef. It's like Whoever's Pizza versus Whoever's Pizza in some city.
I guess a villain kind of reflects some of the flaws in a hero, so Joker pushes the lines of decency like an edgy comedian and Batman just won't kill him. His antipathy tests the heroism of the no kill rule. So on so forth and after that it's basically just a small-town rivalry between egos, because they're both criminals, deviants and immoral, it just depends who is casting the aspersions.
In Dick Grayson's case it's hard. He's very optimistic. So you might think, oh give him a cynic! They rolled that out with James Gordon Jr. in Snyder's Black Mirror. Good run of comics, but that dude as a villain does not have legs. Plus you can't make anyone "Gordon" into a Grayson rival. But more than that ... he's optimistic but dude is not naive or idealistic. He grew up in Batman's house. He's shrewd.
They tried the "we were punk kids together thing" a dozen times, but foremost with like, well, Crime School for Boys, but more recently, Shrike. No legs. Then again he had a pretty bad late 90s costume. Real bad costume. Plus everyone was just assassins back then in the late 90s.
They've tried the femme fatale angle a few times and it never seems to work but probably because it seems childish and rote? Dick has a string of lovers (more or less all fellow superheroes) that are all such good people (and cool characters) that we constantly debate who is the best one for him like a celebrity couple that broke up! The answer of course is that they're ALL really good for him. The common denominator is Dick Grayson. He's bad for all of them. Dick Grayson is a flake.
How do you make a femme fatale expoit THAT though? He actually IS dating a superhero ... but then flakes on her even after there was marriage teases ... he pulls a Kory (or Babs) and leaves her at the altar or close to it, or I mean, he could cheat. Dude lives life keeping secrets ... and THAT is what causes her to become a super-villain. That mister perfect Dick Grayson, god that he is, doesn't want her? Kind of weak. Oh I mean it happens to people in real life, they become enemies because of heart-break. I just think you know, it's a weakness. It's not a healthy approach. Which of course, screams super-villain to you? Well of course. But not a STRONG arch-nemesis of a super-villain who can match Dick in wits, blows, and emotional headgames. That's a nemesis but not an ARCH-nemesis.
So man, what's the approach?
The easy answer is of course ... if Dick Grayson is a spy, his opposite number in the rival organization instantly gets to be a hand-crafted arch-nemesis. I mean, you could even make him go villain and it could actually be NEMESIS. A lover boy, a pretty boy. A master spy. Dick might be more the fighter, but Nemesis is the master of disguise. Dick dated Koriander? Nemesis dated Wonder Woman. And now he's a villain. But a rival with a code. Just not Dick's code. Not Batman's naive super-hero code. For the greater good, people gotta die. And Dick Grayson is in the way. And maybe just maybe it's the moral argument. The logical argument. And what's more ... in his youth ... Alfred Pennyworth, Dick's fatherly surrogate figure who he loves and respects ... used to be a Nemesis type guy. An agent, man of action ... and killer, which pits the ideals and values of BOTH of Dick's role models against each other in his mind.