In the first Deadpool film, Wade Wilson held a deadly grudge against Ajax, aka Francis Freeman. His humorous yet deeply personal vendetta played a central role in the film. And naturally, the film's rivalry between Deadpool and Ajax was heavily inspired by their history in the comics.
Ajax first appeared in 1998's Deadpool #14, by Joe Kelly and Walter McDaniel. In his first few appearances, Ajax was seen hunting several of Wade's fellow Weapon X subjects. It was unclear exactly who Ajax was, outside of his desire to kill Deadpool and his connection to Weapon X. Eventually, Ajax captured Killebrew, the doctor in charge of the Weapon X project when Wade became Deadpool.
Ajax tortured Killebrew until he got Wade's teleportation frequency, allowing him to teleport Deadpool to the Swiss Alps where Ajax punched him off a cliff. During Wade's near-death experience, Death helped him look back on his time at Weapon X.
During Deadpool and Death Annual #1, by Joe Kelly and Steve Harris, Ajax was revealed as an enforcer for the Weapon X project, keeping Wade and his fellow test subjects in line. Ajax was a cruel, sadistic man who literally felt no pain. In addition, Ajax's strength and speed were enhanced.
Wade wanted to die and provoked Ajax to kill him. When Wade learned that Ajax's name was Francis Fanny, he showed no mercy in his mockery. Wade's comments were relentless, encouraging the other Weapon X subjects and angering Francis to no end. Ajax was sorely tempted to kill Wade, but Doctor Killebrew wouldn't allow this. Instead, Ajax nearly killed Wade's friend, Worm. In Worm's broken state, Wade was forced to mercy kill him.
After this, Killebrew authorized Ajax to kill Wade. When Ajax appeared to murder Wade, however, Wade's healing factor jumpstarted. This process allowed Wade to break free of his restraints and destroy the Weapon X facility, seemingly killing Francis as well. Of course, Ajax was still alive, hunting Wade and Doctor Killebrew in the Swiss Alps.
Ajax chased Deadpool, Killebrew and Wade's new friend, Ilaney Brukner, through the Alps, giving Wade quite the challenge. Wade was conflicted about killing Ajax, even as the ghosts of his former Weapon X friends told him to do so. Later, Killebrew sacrificed his life so that Deadpool and Ilaney could escape. Resolving to end the battle once and for all, Deadpool set a trap for Ajax and killed him.
Yet this wasn't quite the end for Ajax, who returned in Deadpool vs. Thanos #2, by Tim Seeley and Elmo Bondoc. When Death went missing, Thanos and Deadpool went to search for her. Along the way, Deadpool was confronted by Ajax, who had been sent from Hell as the Abyss Man. Thanos tortured Francis for Death's location, being led to believe she was in Mephisto's realm. Ajax died once again when he was used in a ritual to open a portal into Mephisto's realm.
Overall, Ajax in the comics was pretty similar to Ajax in the first Deadpool movie. Both versions were sadistic, ruthless killers who figuratively and literally felt no pain. Ajax helped create Deadpool and became Wade Wilson's tormentor in both incarnations of the story.
There is one perculiar difference, however. In the comics, Ajax went out of his way to hunt down Deadpool, while the film saw Deadpool relentlessly searching for Ajax. This is probably because Ajax nearly killed Wade in the movie, while Deadpool nearly killed Ajax in the comics, leading both parties to seek revenge in different media. Another aspect of Ajax which was adapted very well into the film was his real name, Francis.
Both the comics and the film featured Wade mercilessly taunting Francis. Ajax was also very sensitive about his name in both versions. Still, Ajax's job in the film was quite different from the comics. Ajax's movie counterpart was the doctor at the Weapon X project, while the original Ajax was an enforcer for the project. Even so, both versions greatly enjoyed torturing Deadpool and his fellow test subjects.