Please Note: In order to access the paid DLC; "Record's Fragment: Keian Command Championship," you MUST play the base game until you reach Chapter 2 "An Exchange of Vows" where you will be able to leave Akasaka.

Story Completion Screen: The story completion percentage screen is available to view after you've completed at least one full playthrough of the base game. The DLC percentages can be toggled by pressing the respective button on the bottom corner of the completion screen.

Screenshots will be from the Nintendo Switch version of Fate/Samurai Remnant.
The content and gameplay across all three available versions are the SAME.
All available versions: Nintendo Switch, PlayStation and Steam.

Following the opening dialogues, you will be taken to the menu where you will have access to ONLY three of seven teams. Those teams will be Team Saber, Team Rider and Team Berserker. As you complete each team's available story, you will slowly unlock the others. The order you unlock the other teams will vary based on the order you decide to complete them. This also means the order in which you complete them does NOT matter as you will likely have all teams unlocked after completing your 6th team's playthrough.
Note: For my playthrough, I tried to keep it in order from left to right starting with Team Saber. This resulted in Team Assassin being unlocked last for me.

After selecting a team, you will be greeted by "Boss" who will offer to explain the rules and reasons behind his event. Part of the completion process is listening to all of Boss' explanation. Thankfully, you only need to do this once with a single team. On all other subsequent playthroughs, you can forego his explanations without impacting your completion.

The first mission of every team (except Caster) will be to find and collect the golden Boss statues. In total, you will collect 7 of them, one for each team you playthrough. These will be important for full completion. Once you find and collect the statue, each team will be forced to face a rival team to complete the mission. Once defeated, you will be taken to Boss' special arena where you will be told the statue is yours to keep. For Team Caster, you must BUY it from the shop keeper using Boss' unique currency. You will earn the DLC exclusive currency as you battle.

Boss will then hint to you that there is another use for the statues, and your character will encourage you to explore the resting area to figure out the other use.

Each of the golden statues have a unique golden pedestal they must be placed upon. The location of them vary, and may involve requiring you to climb up or down to find them. Some pedestals may appear to be blocked off, but that simply indicates that it is the incorrect pedestal for your particular statue. The area isn't big, so it'll be difficult NOT to find the correct one. You will know if you've found the correct golden pedestal if you're able to walk directly next to it and place it down.

After placing down the statue, your characters will react as though it is not important but this will play a huge role later so make sure you're doing this with each team. For now, complete the remaining battles to finish the route.

Remember when I mentioned the importance of the Golden Boss Statues? While completing your 7th team, once you place down the 7th (and last) Golden Boss statue, you will be taken immediately to a special arena to face a slightly modified battle against Boss himself. The battle is difficult, and will require proper usage of servants. Thankfully for this battle, the crowd gauge will be present at the top corner, allowing you to gain the same buffs as you've had when facing the other battles.

After you defeat boss, a unique dialogue will play with the team you used. You will then be taken back to the main arena where you can complete the remainder of the team's story.

Once you finish all the team routes once, you may have noticed that the completion percentage is at 94%. If you are less than 94%, you may have either not finished a team route yet or have not listened to Gilgamesh's explanations.

At this point, the remaining 6% will involve you replaying each team's route to place down the Golden Boss statue again and then fighting him in the special arena. You only need to fight him once per team. If you already fought his special arena with your 7th team, you do NOT need to face him again.

Once you complete the "Boss" battle with a team and return to the rest area at the arena, you are welcome to pause and exit the DLC. Despite the warning message, your progression WILL be counted towards the overall completion record, and will increment by 1% for each team you've completed the battle.

I hope this guide helps you reach full 100% completion for Fate/Samurai Remnant First DLC Pack - Record's Fragment: Keian Command Championship. If you encounter any issues/or have any questions, you're welcome to ask them below and I'll do my best to assist you.