Immortality is one of the central themes in Hell's Paradise: Jigokuraku, and the series MacGuffin, the Elixir of Life (now known as "Tan") is an in-universe means of achieving eternal life. The inhuman abilities developed by the Lord Tensen, the series' antagonists, are one part due to their vigorous training in the Five Training Methods to Immortality which developed their mastery over the mysterious power known as Tao; while another part of their overwhelming might is due to their consistent consumption of Tan.
Even though the Tensen have presided over the elusive secret to eternal life for literally centuries, a member of the Vanguard Party figures he could become immortal through a method the Tensen do not know about: fame. Gantetsusai Tamiya, known by his monikers "Blade Dragon" and "Eight Provinces Unparalleled" reveals his motivations for participating in the search, and unlike his contemporaries, the pardon has nothing to do with it.
Dragon Slayer
Gantetsusai's character is constantly compared with or linked to dragons. His skill with the blade is definitely the main reason behind this constant juxtaposition; however, the dragon is the character's most important motif because it has guided him throughout his arc. Having been trained in swordsmanship from an early age, Gantetsusai gained a reputation for his strength in combat, which was fabled to be able to best even that of a dragon. In his recollection of the reason for his arrest and subsequent placement on death row in episode 4, Gantetsusai's boldness and stubborn personality are evident because of how trivial the circumstances are. Gantetsusai, having been recognized for his strength, became the drinking buddy of a local lord who doubted that he could actually slay a dragon as his epithet claims. Gantetsusai proved his strength through metaphor by cutting the torii gate leading to the lord's home in half. This gate bore the crest of a dragon within its arch, and through its destruction, Gantetsusai was addressing the lord's claim directly. This had him arrested and subsequently placed on death row, leading up to his involvement in the expedition to Shinsenkyō.
Dragons are interesting mythological creatures for their universal prevalence in human history. For whatever reason, societies across time and space have conceived of gigantic lizard-like creatures that possess keen intelligence, and malice towards humanity. These creatures are often the ultimate depiction of fearsome strength and terror, and the fact that they exist in virtually all mythologies paints them almost as humanity's collective ultimate atrocity – as humanity's imagined supernatural predator. Gantetsusai's reputation characterizing him as an individual who can not only stand against, but defeat creatures that haunt our collective nightmares sets him up to be someone who is overwhelmingly strong.
Go Big, Or Go Home
Gantetsusai has an incredibly strong personality. His characterization is not unlike many swordsmen in anime like the BLEACH series' Kenpachi Zaraki, in that Gantetsusai is extremely motivated by combat and the prospect of encountering strong opponents. Overall, Gantetsusai's very character is supposed to exude strength, and it is also for that reason that the true horror of the island was first shown through him. Gantetsusai is the kind of character who wouldn't hesitate to chop off his own hand if the situation called for it, and his quick thinking in such a situation shows that he possesses a decent enough level of intelligence to be able to process how to react to a given situation.
Even with only one hand, Gantetsusai has survived on the island long enough to encounter Gabimaru and the crew, and still felt confident enough to challenge the ninja to a fight. In this encounter, Gantetsusai reveals his reasons for participating in the search for the Elixir of Life, which have nothing to do with the possibility of being pardoned for his crimes. In fact, Gantetsusai couldn't care less about any of it – his motivation for being on Shinsenkyō is directly related to his big personality. As someone famed for his combat ability and swordsmanship, Gantetsusai is someone whose reputation means everything to him, and it is out of a desire to protect his ego that he ended up on death row in the first place.
Dragon Slayer (of Legend)
Gantetsusai's characterization being "the strong guy" goes beyond his strength as his entire being is dictated by strength as vice and virtue. Because he is strong, he is feared, but when someone didn't react to his strength, it led to him getting himself into trouble without care for consequence. All Gantetsusai wants is to be recognized for being "the strong guy", and his strangeness comes through when he talks about not caring about immortality in the literal sense at all. If Gantetsusai found the Elixir of Life, he would simply grab it and go home, because he desires immortality in a different context to literally having eternal life. He wants to be the legendary warrior who found the fabled elixir, not someone who merely drank it. As such, Gantetsusai has figured out a method to immortality that the Tensen have never considered: overwhelming, indisputable glory.