Leary is a sad, psychotic loner who used to be an assassin, until he was ‘betrayed’ by the US government who cut him loose. Presumably he said or did some troubling things which led to the CIA sending a former colleague/friend of his to kill him, he reacted by slashing his friend’s throat.
So, he’s extremely bitter and enraged and wants to get back at the US government by assassinating the President, making him and his story famous.
He notices that Frank Horrigan, who famously failed to protect Kennedy, is still in the game. Lonely Leary feels a kinship with Frank because he sees both men as screwed over by the system and living alone. This is why he ‘saves’ Frank on the rooftop.
He wants to persuade Frank to see the world as he does, but because he’s a sicko, he also enjoys taunting Frank and making his mission to kill the President a ‘game’ that the two men are playing. So again, he needs Frank alive to play his game.
It’s a complex and well-written villain, perfectly played by mad Malkovich.

He related to Frank as someone who also got screwed by the government/system like he (supposedly) did.

Because it makes it more enjoyable and worthy for him, going against the guy who was head of Kennedy's security detail. He wants to be rememered for what he's planning to do.

You have a point. Also do you think that Frank did the right thing that he didn't allow Al to resign?

"Why did he want an adversary?"
The short answer is that Mitch Leary's character is insane or mentally unbalanced.
The long answer is that he sees his assassination plot as a "game" of sorts and he wants to be challenged by an equal player in this "game". Leary has no political motivations whatsoever. Consider Leary's own words on the subject
Mitch Leary:
There's no cause left worth fighting for, Frank. All we have is the game. I'm on offense, you're on defense.
Mitch Leary:
It's random and it's meaningless.
Frank Horrigan:
Well, if none of this means anything why kill the President?
Mitch Leary:
To punctuate the dreariness.

Why did he want an adversary?

"If he just let Frank die, he would be able to kill the president without any interference."
Yeah and if he wasn't constantly calling Frank on the phone, taunting him, and dropping hints about himself throughout the movie (including WHERE the assassination would takes place) then he would be able to kill the pres. "without any interference" as you say.
But Mitch didn't want to simply commit an assassination without interference. He wanted to play a psychological game and to go up against an equal opponent. Think back to the phone call in first act of the movie
MITCH: "Trying to trace me Frank?"
FRANK: "Now why didn't I think of that?"
MITCH: "You did. Or are you not the adversary I had hoped for?"