Frank is usually treated as an absolute moralist who just happens to be fighting a never ending war on crime. Other interpretations imply he's just as bad as the monsters he fights and is little different from a serial killer. The only thing that stays the same is the reason why he became the Punisher.
Punisher: Year One has yet another take on Frank's character - he goes vigilante only after all other methods of bringing the mobsters who killed his family to justice fail spectacularly (the police are ineffective and corrupt; an attempt to do an independent investigation leads only to the death of the reporter whom Frank briefly befriended).
Punisher: Born from MAX imprint has a different take on WHY he became the Punisher. Frank fell in love with killing and made a deal with an unknown supernatural entity (or just plainly went mad, you choose) to fight a war that would last forever. He wasn't informed until later that the price would be his family, at which point the supernatural entity makes Frank forget that they ever made the deal.
In The Punisher: Purgatory, Frank was turned from a mafia-hunting anti-hero into a supernatural force of vengeance on a path to redemption to be able to rejoin his family in heaven. This was undone with a Handwave by later writer Garth Ennis in his "Welcome Back, Frank" storyline. Despite the handwave, a subtle nod to the Heavenly Enforcer period appeared in a volume two Thunderbolts issue, after Deadpool acquires an angel feather while in Mephisto's hell it heals Punisher from fatal injuries he'd received while he was still on Earth. Deadpool asks how someone so non-Heavenly as Punisher could possibly have anyone in Heaven interested in him (to paraphrase) only for the Punisher to blow him off with a "don't want to talk about it".
Likewise Franken-Castle will be remembered fondly as a brief period of lunacy in Frank's life. There's no way the powers that be intended for a magic/SCIENCE half-robot Frankenstein's Monster Punisher to be a new cutting edge status quo. Even in the Heroic Age. These days Franken-Castle exists only as an AU Punisher from a Monster Mash universe. He is part of the Avengers there (which also include monster forms of other heroes who had them in the mainstream 'verse in the past, like werewolf Captain America - see below - and vampire Wolverine). They only appeared briefly to tangle with the regular New Avengers and were offed unceremoniously.
John Ostrander's oft-forgotten run had Frank become the head of a Mafia family before things turned into an X-Men story that happened to star the Punisher, with Frank being roped into helping X-Cutioner and S.H.I.E.L.D. rescue an activist from the latest incarnation of the Mutant Liberation Front (here a false flag group backed by Humanity's Last Stand). The final issue had Frank incur amnesia, leading to the aforementioned Purgatory.
Frank Castle inexplicably joining Hydra in Secret Empire has been met with nothing but scorn by Punisher fans. Some believe that Nick Spencer only did this to anger Punisher fans.
Harsher in Hindsight: In one Punisher comic, Frank confronts a group of police officers who have taken to displaying his skull logo on their vehicles, indicating that they intend to take his approach to dealing with crime. Frank is...less than pleased.◊ In 2019, the St. Louis Police Union instructed its officers to begin displaying a variation of the "Thin Blue Line" flag superimposed over the Punisher's logo, in solidarity with officers under investigation for having adopted the same logo on social media.
Frank and Microchip would come off like an old married couple sometimes. In some issues, Micro would even use the word "bubbelah" (Yiddish for "darling") when talking to Frank.
Punisher: Up for thirty-six hours watching a crack lab, and Micro calls me with a grocery list on the way home. Sometimes having Microchip is like having a wife.
The most overt instance of this probably the first issue of War Zone, where Frank and Micro bickering over Micro sneaking off to see a therapist comes off like a couple cracking up due to one member uncovering another's infidelity.
Microchip: ¢%$#! I don't need this, I'm going out.
Punisher: Where? I might need you later.
Microchip: Just out. *¢%, we're not married, Frank.
Franken-Castle's fight with Daken was filled with homoerotic banter.
Magnificent Bastard: Frank himself. Agree with his methods or not, but either way he is damn good in what he does. Just try to keep tally of the times where he got out of a seemingly impossible situation or pulled off a complicated plan to get to his target. Occasionally he even outsmarts super-powered beings far more mighty than him. What Reed Richards is to Science and Tony Stark to Mechanics, Frank Castle is to Warfare and Survival.
Misaimed Fandom: Gerry Conway has stated he's uncomfortable with how popular the Punisher is with members of the American military and law-enforcement, as he is very anti-war and anti-authoritarian as well as missing the point of a vigilante. He intended for the Punisher to be a tragic anti-hero with ambiguous and sympathetic traits but as primarily an example of the failure of America's military and legal systems:
Gerry Conway: "The vigilante anti-hero is fundamentally a critique of the justice system, an example of social failure, so when cops put Punisher skulls on their cars or members of the military wear Punisher skull patches, they're basically sides with an enemy of the system. They are embracing an outlaw mentality. Whether you think the Punisher is justified or not, whether you admire his code of ethics, he is an outlaw. He is a criminal. Police should not be embracing a criminal as their symbol...My point of view is, the Punisher is an anti-hero, someone we might root for while remembering he's also an outlaw and criminal. If an officer of the law, representing the justice system puts a criminal's symbol on his police car, or shares challenge coins honoring a criminal he or she is making a very ill-advised statement about their understanding of the law."
An in-universe version happens when Frank meets people who actually did put his skull on their car and rips it up, telling them that if they want to help, Captain America is always around, but nobody should be like him (see Harsher in Hindsight above).
Sequel Displacement: While Frank was one of Marvel's cash cows back in the early nineties (alongside Spider-Man and Wolverine), many of his modern fans know nothing about his stories before Ennis. Sadly this means that a lot of his gentler characterizations are overlooked, as well as the fact that he actually did have something of a supporting cast.
Strawman Has a Point: The liberal social worker who Frank meets in one storyline points out that Punisher kills society's victims more often than he kills true villains. He even acknowledges that she has a point when he later confronts a group of murderous homeless and find it difficult to pull the trigger, since most of them are too far gone to understand what they're doing anymore. In an uncharacteristic move, he gives them all a chance to flee, a chance most of them take.
Complete Monster: General Kreigkopf is the leader of the Russian Syndicate in the second portion of the game. An associate of the Gnucci Family, Kreigkopf would supply them with drugs and money from the harbor. When Castle attacks the harbor, he discovers Kreigkopf has been smuggling in weapons, including a tank and a nuke. Frank discovers that Kreigkopf is also part of a Human Trafficking ring. Kreigkopf sends the Russian to attack Castle at his apartment. When Castle and Fury invade Grand Nixon Island, he prepares to launch the nuke at New York City, and manages to get it to the point it's supposedly irreversible before Frank kills him.