Hector Bautista was a convicted murderer sitting on death row in Texas. One night, the Time Stone (having been given life by Adam Warlock using the Soul Stone) broke into his jail cell and possessed him. Hector is able to use his new found ability to start and stop time to escape.
Overtime debuted in Infinity Wars: Infinity #1 by writer Gerry Duggan and artist Mark Bagley. He was a spinoff of the Infinity Wars event, which ended with Adam Warlock imbuing the Infinity Gems with the ability to choose their own destinies. He was introduced ahead of the Infinite Destinies event that would chronicle the Gems' new chosen ones, but Infinite Destinies was delayed due to the global COVID-19 pandemic, leaving a decent sized gap between his first appearance and the publishing of the event.
Honing His New Abilities
Hector is not only on the run from the cops. He has also been tracked by Warbringer and the Fraternity of Raptors, led by Talonar. Although he evaded Warbringer, the Raptors were able to kidnap him and bring him to space. They strapped him into a torture device in order to figure out how he manipulates time. Luckily for Hector, Wolverine and Loki have been charged with protecting him by a version of Wolverine from the future. So while Wolverine hacks and slashes his way through Raptors, Loki tries to psychologically manipulate Talonar into letting Hector go.
Loki's plan doesn't go exactly as planned. He teleports Talonar's mother on to the ship, but Talonar, thinking she was an illusion, killed her. Loki is able to free Hector, but he talks Hector through time travelling backwards to save her. At this point, Hector had only figured out how to start and stop time. Once he figured it out, the older version of him faded as a paradox, and Talonar's mom was safe. Unfortunately, Warbringer showed up at that exact moment.
After a pep talk from Wolverine, Hector stops time entirely and gets to work practicing rewinding and fast forwarding time. As soon as he is confident in his powers, he hits the play button and stops just Warbringer. This gives Talonar the opportunity to teleport everyone back to Earth. This is when Wolverine starts demanding Hector help him figure out how to save the X-Men since Wolverine found the Academy in disrepair. Hector doesn't like his attitude, so he quickly hits him with as many punches as he can and then disappears.
After saving some folks from a fire, Hector ended up on Black Widow's radar. She suspected he used the fire to pickpocket all the people he rescued. So, she and Captain America went to investigate.
Unfortunately, Widow and Cap were antagonistic toward Hector from the beginning. They cared more about his murder conviction than any of his good deeds since getting the Time Gem. Hector decided to run, using his time manipulation to evade their many attacks, believing that with more practice, he'll be able to travel back in time and stop himself from ever murdering those shopkeepers.
With the Time Gem, Overtime is able to start and stop time on a whim.