X-Factor has changed a lot through the years and the version that works in Krakoa right now is nothing like the version that first broke onto the scene in the 1980s. X-Factor debuted in 1986 from the creative team of Bob Layton and Jackson Guice as a way for Marvel to profit from fan nostalgia to reunite the original X-Men once again.
That team had one purpose: to find scared mutants and saved them from humans who wanted to hurt them. That changed over the years, with Havok's team becoming a new government mutant task force and then becoming a group that helped investigate deaths and other mutant crimes in Krakoa.
Jean Grey
The formation of X-Factor occurred because Jean Grey returned from the dead. In the first (and one of the best) issue of the X-Factor comics, Reed Richards of the Fantastic Four had called in Angel and told him they found Jean in a cocoon in a harbor and the Dark Phoenix that died was only posing as her. This led to Jean and Angel bringing the original X-Men members back together.
What is important here is that it was Jean Grey who demanded that the original X-Men reform when she saw how the hatred and intolerance among humans had risen to unbearable and dangerous levels for young mutants. Jean's demands to help mutants forced her former teammates to relent and join her cause.
Angel
Angel was the first person to learn that Jean Grey was alive and went to talk to her before contacting the rest of the original X-Men members. While Cyclops wanted to keep a distance and originally rejected Jean's ideas, Angel sought her out when she left and agreed that they should form this new team.
It was also Angel that was instrumental in establishing X-Factor as convinced his friend Cameron Hodge to help the team pose as mutant hunters but were really saving these persecuted characters and helping train them. Angel was also the most tragic member, as he lost his wings during his time with the group and returned as the deadly Archangel thanks to Apocalypse.
Beast
The first person that Angel and Jean Grey convinced to join the team was the Beast. It was perfect timing as well. Angel, Beast, and Iceman were part of the Defenders and that group had just broken up. Beast was trying to get a job at a university, but they rejected him because he was a mutant.
That is when he got the call from Angel and when he showed up and learned Jean was alive, he was quick to join. Beast had an interesting time in X-Factor, turning into his human form again and losing his now-signature blue fur for a time, which helped him blend in when they pretended to be mutant hunters.
Iceman
Iceman joined up at the same time as Beast, but his story played out just one page later when he was working a boring office job and jumped at the chance to get back into the action. As the youngest member of the original X-Men, Iceman was still ready to be a hero.
His stay with the team went better than most. He never had the tragic narrative arc of Angel or the back-and-forth transformations of Beast, while also remaining the heart of the team, a young mutant wanting to help others.
Cyclops
Cyclops was the first leader of X-Factor, but he was also the last to join. With Jean Grey back from the dead, Cyclops was put into a tough spot. He married Madelyne Pryor after Jean died and the two had a son named Nathan. He left his wife and baby to join X-Factor, one of Marvel's best non-X-Men mutant teams.
This caused him a lot of pain at first, and his relationship with Jean was strained because he hid his wife and child from her. By the time he returned home, Madelyne and Nathan were gone. Scott joining X-Factor led to several major events, including his wife becoming the Goblin Queen and his son becoming Cable.
Havok
Over the years, X-Factor saved several young mutants from humans who hated and feated them. These mutants became known as X-Terminators and helped X-Factor on their missions. However, when X-Factor broke up and X-Terminators merged with the New Mutants, the government wanted to form a new X-Factor team.
Government agent Val Cooper had just watched Freedom Force disband. She decided to start a new government-sponsored mutant team and wanted a respected mutant in charge, which led her to choose Scott Summers' younger brother Havok. It took some convincing, but she finally brought him on and he became the new leader. Alex would largely stay with the team until the series ended and was rebooted as Mutant X, an alternate reality comic series that had Alex living another life as the lover of Madelyn Pryor.
Wolfsbane
When Val went to find Havok, he happened to have a close friend with him at the time in Wolfsbane. This is Rahne Sinclair, one of the founding members of the New Mutants. When Havok joined up, Wolfsbane joined him, showing an attraction that was part of the remnants of her bonding with him forcibly on Genosha.
Wolfsbane remains better known for her time with the New Mutants since she is currently back with that team helping train the young mutants on Krakoa. However, she rejoined X-Factor two more times thanks to her friendship with Jamie Madrox, who was close to her thanks to their shared time with Moira MacTaggert on Muir Isle.
Polaris
Havok refused to have anything to do with the team at first, but then Val mentioned that Polaris was going to be part of the team, which brought in Havok instantly. Polaris and Havok had a long history together and a romantic relationship in the past that gave X-Factor something extra when the two joined forces again.
Polaris ended up as a co-leader with Havok and lasted even longer than he did, as one of the core members of the mutant superteam. When X-Factor reformed in Krakoa, she took over as a co-leader with Northstar.
Strong Guy
Along with Polaris, Strong Guy was another mutant that Val brought in to serve on the team. Before this, Strong Guy was a new mutant in Marvel Comics, one of the mutants found living on Muir Island and controlled by Shadow King.
With his rescue and the defeat of X-Factor, Strong Guy accepted Val's invitation to join the new team along with Polaris. He also provided humor for the group and he came up with his superhero codename at this time based on a joke at their introductory press conference. Strong Guy was one of the most powerful members of the New Mutants but is best known for his time with X-Factor.
Jamie Madrox
While Cyclops and Havok were the first two leaders of X-Factor, it was Jamie Madrox as Multiple Man who took the team on his shoulders and changed what it meant to the mutant world. Jamie joined at the same time as Polaris and Strong Guy, under Havok's leadership, but he ended up taking over as the leader in 2004.
Madrox is who turned X-Factor into a private investigator business. He licensed the name and opened up X-Factor Investigations with Strong Guy and Wolfsbane working for him. The PI aspect is what X-Factor remains known for to this day.