Heroes who usually go against villains who are Lawful Evil (the ones who use the law for their own purposes and evil deeds). Sometimes may commit crimes such as murder, theft, terrorism, vandalism, etc. (Juvenile Delinquents and Outlaws count), but that still help save the day. However, often the people that these heroes look after don't show any gratitude, because the law system and society probably paints these heroes as villains via propaganda. There are several reasons why a good person would resort to crime. 

• Vigilante: The society/law system they live in is corrupt and oppressive and is making things extremely difficult or impossible for a good or neutral person to make an honest living. Or the system itself is the source of conflict of the story via the injustice it inflicts, forcing heroes to resort to crime and/or vigilantism to survive and correct wrongdoing. This is the most common form of a Heroic Criminal and in some ways the rest of the examples fall under it. Examples include Zorro, Aladdin, and Robin Hood.
• Revenge: The hero experiences a personal tragedy involving one or more members of his family and/or other loved ones (often with fatal consequences) and due to the inability (or unwillingness) of the law system to properly prosecute and punish the criminals who wronged him and the ones he loves, he ultimately chooses to take matters into his own hands (i.e. vigilantism), acting outside of the law in order to make justice and avenge his loved ones and other innocents. Examples include Frank Castle/Punisher, Mack Bolan, Carrie White, Paul Kersey and Kim Soo-hyun.
• Minority Species/Culture: The hero is a member of a species, culture, religion, sex or gender, or any group that is frowned upon and/or discriminated against by a majority species/culture, forcing them to resort to crime in order to survive since the local law enforcement probably is biased against their species or culture and the society likely denies them rights and good wellfare. Common in Science Fiction and Fantasy where multiple races can be found, and related to real life racism, the hero resorts to crime because the local law enforcement is more likely to arrest them rather than help them, simply because of their species or culture. Examples include the Dovahkiin if they are an Argonian or Khajiit or Esmeralda for being a gypsy. 
• Poverty: The hero resorts to crime out of poverty and survival. Often the society these heroes live in care more about money, power, and the elite more than commoners (and often these commoners are among a disadvantaged group) and does anything to ensure that poor people "know their place". Good examples include Catwoman and Aladdin.
• Mafia/Gangsters: These heroes are rare type of being a criminal. They may be ruthless, but they have good intentions in mind. Likewise, heroic gangsters all have a deep trust and respect in each other, and their ultimate goal is to take over other mafia and turn it into a force of good for the people of their homeland. Examples include Team Bucciarati and Jackie Estacado
• Being Framed: The hero is actually a decent law abiding person, until they were framed for a major crime by a villainous character and are now a wanted criminal. This character is typically found in mystery stories where the hero tries to find out who really committed the crime so they can clear their name. But until then, they must resort to crime since law enforcement is more likely to try and arrest them than listen to them. Examples include Dr. Richard Kimble, Judah Ben-Hur and Vash the Stampede.
• Tragic History: Sometimes the hero becomes a criminal because of their tragic backstory like being an orphan or being rejected by society so they had no choice but to become a criminal. Examples include Megamind, Catwoman, and Gru.
• Anti-Hero: The hero simply has a tendency towards criminal behavior, but is still ultimately a heroic person. Examples include Duncan.

Heroic Criminals can qualify as Pure Good, but they can ONLY qualify if their crimes are for the sake of others and not for their own needs.