One important element could be their motivation. If Shura is a character in a story about revenge, then something bad that happened to them or their loved ones in the past is a key part of the back story. It could also be a goal, like Shura aiming to become the best in a certain field, which drives their actions and shapes their back story. Another aspect could be relationships, for example, a mentor who influenced Shura's growth.
As mentioned before, not knowing the specific Shura makes it hard. But generally, key elements could include their origin, like where they were born. Family might be another, whether they had a loving family or a troubled one.
As I don't know the specific 'Shura Kirigakure' story, one possible key element could be his origin. Where he came from within the Kirigakure setting would likely be important.
One key element is his origin as a Maia. Another is his association with Morgoth, which shaped his evil nature. Also, the creation of the One Ring is a crucial part of his back story.
The key elements are his cancer diagnosis, the secret experiment he joined, and his resulting superhuman healing factor.
One key element is his fall into the vat of chemicals. It was a turning point that physically and mentally transformed him.
Family or lack of it often plays a big role. Maybe his family was killed by criminals, which drove him to become a crime - fighter. This would be a significant part of his back story, as it explains his actions and his alliance with others in the fight against evil.
One key element is his sense of justice. He saw the wrongs done to kids and wanted to fix them.
One key element is his upbringing in a wealthy family which allowed him to pursue his passion for hunting. Another is his boredom with normal hunting and his consequent search for a more challenging quarry, which led him to Spider - Man. Also, his sense of self - importance and the need to prove himself as the greatest hunter are important parts of his back story.
One key element is his thirst for strong opponents. It's what drives him in the story.
One key element is the influence of different musical styles. As I mentioned before, it draws from rock, R&B, etc. Another is the role of media. Radio and later TV were crucial in making pop music popular. For example, shows like American Bandstand introduced new pop acts to a large audience.
One key element is his scientific genius. It's what got him into all the multiverse adventures. Another is his family. His relationship with his daughter and the loss he experienced within his family are major parts. Also, his recklessness in his experiments which led to so many consequences both good and bad in his life.