The Punisher series is moderately close to the comic. Some key elements and storylines are retained, but there are also some variations to adapt to the medium of TV or film.
The Punisher, or Frank Castle, was a normal guy with a family. His military background gave him a set of skills that were both useful and deadly. Once his family was massacred, he snapped. He turned his skills towards punishing those who he thought deserved it. His origin story is one of tragedy and a thirst for justice outside the normal bounds. He doesn't wear a colorful costume like some superheroes. Instead, he has a simple skull emblem on his chest, representing death to those who do wrong.
One of the best is 'The Punisher: Born'. It delves deep into Frank Castle's origin story, showing his transformation from a military man to the Punisher. The gritty art and intense storytelling really bring out the essence of the character.
Overall, the Punisher adaptation remains relatively close to the comic book. It keeps the basic plotlines and the tough, no-nonsense image of the character. However, there are some changes made to fit the medium of TV or film and modern audiences.