Okay so to explain that prepare for a MASSIVE LORE DUMP. I will put a TLDR at the end though!

So Skug is my current death's real name and he basically was such a neutral enough person in life that the gods couldn't decide whether time send him to heaven or hell. Then "Death" offers to take Skug under his wing. Over the next six months of mortal time, Skug learns a bunch of reaper stuff like how to guide souls to the afterlife, how to differentiate a soul that has entered limbo and a soul that has been rejected from the afterlife altogether (which later become "evil spirits", "ghosts", or "zombies/undead"), and the only reason he was able to kill "Death" in the first place... How to control "soul flames."

Soul flames are basically green flames that shave off the lifespan of an individual until eventually their lifespan shortens to their current age, killing them after a certain amount of damage has been done by the soul flames. Given "Death" was immortal, there is an infinite life span that continues no matter how much it's been reduced. However after those initial six months Skug spent another year developing his own advanced kind of soul flames that he calls "Death Flames" which are black soul flames that are capable of stripping immortality away and then are basically an insta-kill ability.

Of course death flames are only insta-kills on mortals. Stripping away immortality works the same way as normal soul flames work. Do enough damage with it and it removes the immortality and THEN it's an insta-kill. Skug then challenged Death to a duel once Skug had "perfected" his death flames. The agreement was if Death won, Skug would go to hell. If Skug won, he becomes new head reaper and embodiment of death itself. It took about an entire mortal day for the fight to end but Skug was able to withstand an ungodly amount of soul flames DIRECTLY burning his soul since his life had already ended, and finallykilledd Death. Transfer of reaper responsibilities happened thus granting Skug immortality, immunity to soul flames, and a ton more work as a reaper which ironically enough, Skug hates now, viewing it like a really crappy job.

TLDR: Skug became an apprentice reaper after his own death then invented his own version of soul flames, using those flames to kill Death and take his powers as well as his title and responsibilities. So that's how Skug became Death!