Sword Art Online is one of the most influential series in the isekai genre. Over the course of Kirito's virtual adventures, he comes face-to-face with monsters that could risk his life if he was ever pushed to his limits. Having more knowledge then most of the players who originally got trapped in SAO and gaining enough experience from that to give him extra advantages in the Fairy Dance Arc, it's fair to say that the famed Black Swordsman shined when it came to the battlefield. While his romance with Asuna was sometimes pushed to the sidelines, Kirito's relationship with Asuna serves as a catalyst for many of his goals.
When they first meet, Kirito attempts to isolate himself when it comes to developing relationships with people. For example, he denies the opportunity to party up with Klein while they were stuck on the first floor. In a similar sense, Kirito's relationship with Asuna was strictly business. At the request of the boss raid coordinators that wanted people to pair up, he ends up noticing Asuna alone and decides to work with her for that first boss fight. Asuna also kept a lot of things to herself when they were first trapped in the game. She wore a cloak to conceal her body until the time that Kirito and Asuna fought against the floor boss, with the hood falling off during their battle.
After this fight, Kirito willingly isolates himself from the other players by using his "beater" status to his advantage. He even blows off Asuna despite her attempts to talk with him, giving them a bit of time apart from each other. While their relationship has a sort of friction to it due to their personalities, the two end up growing closer while using their abilities to their advantage. An example of Asuna's usefulness comes when Kirito acquired S-Class meat, something which would be ruined if he tried to cook with it due to his low skill in that regard. Meanwhile, Asuna has high cooking skills. It's because of this circumstance that Kirito shares the food with Asuna, and it's thanks to this interaction that Kirito and Asuna end up forming a party together.
Through their trials and tribulations, Kirito and Asuna fall in love with each other in this virtual world. In fact, the virtual world gives them the chance to live for each other. On top of Kirito and Asuna performing the quest that lets them get married in Aincrad, the two of them get a sort of adopted child in the form of Yui. Yui is an AI who doesn't have all of her memories at first, stumbling into this pair and staying with them at their honeymoon spot. When Yui regains her memory and takes actions that would have her get deleted, the two continue to treat her as if she was their child and Kirito manages to save her.
The Fairy Dance Arc unfortunately detracts from Asuna's role in the franchise due to the fact that she becomes a damsel in distress. While Asuna doesn't get deleted when the game Sword Art Online is completed, she ends up imprisoned in another game world by a person who intends to force himself on her. When Kirito finds that Asuna is in the game he starts his journey to look for her, getting pulled along into some smaller diversions while he inches closer and closer to his real objective.
The movie Sword Art Online: Ordinal Scale gives Asuna a bit of leverage at first when it comes to combat prowess. This movie centers around the use of a new technology that generates virtual images similar to alternate reality games. Because of the fact that Kirito wasn't the fittest person in the real world, he starts off the movie at a disadvantage compared to better players like Asuna. It's thanks to the connections that he made with Asuna and his other old friends that he improves.
It's fair to say that the relationship between Kirito and Asuna isn't the primary objective of the franchise. While Kirito had his growing relationship with Asuna, he struck up connections with other girls including Lisbeth and Silica. While the conflicts of the second season relate to Kirito's past, his experiences in Gun Gale Online aren't directly involved with Asuna. However, despite the directions the story takes, Kirito's bond with Asuna is still an important part of the series' emotional core.