The Thunderbolts was a group of former supervillains that attempted to turn over a new leaf and become heroes. After the Heroes Crisis though, Amanda Waller took over the team's name and turned it into a program. She plans to reactivate it into a functional hero group for U.S. Government purposes.
Baron Zemo looked at the acclaim and acceptance that people gave hero groups like the Fantastic Four and Avengers. They were given all sorts of clearances and could almost act above any suspicion no matter what they were doing. It gave him an idea.
Baron Zemo began recruiting his own team of heroes, starting with Moonstone and Fixer. He filled them in on his plan and convinced them of the brilliance of it. Then, as Citizen V, he started recruiting some flunkies who would help with his cover and be easily manipulated. He chose Goliath (Atlas), Screaming Mimi (Songbird), and The Beetle (MACH-1). They all chose new names and costumes, and Citizen V convinced them they were starting off a new life as Superheroes.
The Thunderbolts appeared on the scene, helping to fight against HYDRA, AIM and other villainous groups, fighting crime and living what seemed to be a very open lifestyle of reformed criminals. Public opinion was mixed about the group, with some untrusting but others willing to believe they could have turned over a new leaf. Early on, there were some misunderstanding clashes with the Avengers and other heroes, but each time it seemed that the Thunderbolts were actually exactly what they seemed to be, heroes.
Only Fixer and Moonstone knew who Citizen V actually was and that while he was helping stop the crimes of others, he was gathering information and intelligence to make himself unstoppable. Not to mention the occasional missing goodie or two that could be blamed on the criminals they were helping catch. Gradually over the months, the Thunderbolts became more and more accepted as a group of heroes.
However, as time moved on, the leader Citizen V announced that he was Baron Zemo, and that they were really the new Masters of Evil. Some of the Thunderbolts really did want to reform however and so with Hawkeye as their leader, they attempted to apprehend Baron Zemo and clear their names.
All of the Thunderbolts were arrested, but Hawkeye and those that followed him at the end were given reduced scentances, and later Presidential Pardons for their attempt at stopping Zemo.
Some time later, after the Heroes Crisis happened, Amanda Waller seized the team's name and built a program. She activates it into a functional hero group for U.S. Government purposes.