No experience with 5e here, but the way iaido it was done back in Ye Olde 3rd Edition Dayes was a combination of a skill called Iaijutsu Focus and a prestige class called Iaijutsu Master. Both of them derived from the 3.0 book Oriental Adventures.

The skill basically entitled you - on the first weapon draw of a combat, which really hobbled/balanced the use of the ability - to make an Iaijutsu Focus skill check. The check result then determined how many bonus d6 dice were added to your damage for that strike, assuming that you hit. The Prestige Class basically then added stuff that should have enhanced the use of that skill, doing things like adding your CHA bonus to initiative, and at the end of the level progression, basically giving the Iaijutsu Master a free surprise round.

Unfortunately, at least in 3rd ed, it didn't turn out to be terribly powerful, mainly because 3.5 options overshadowed it - like Pounce, which allows you to close with an enemy and still deliver a full attack, things like that. Additionally, the bonus damage wasn't bad, but it still suffered from the old Rogue Sneak Attack problem of being very conditional on when it could activate and the damage being nowhere near what a mage or cleric of comparable level could put out. And Iaijutsu Master sweated you for 10 levels for all of its toys, which was generally seen as too high an opportunity cost to really be workable. Indeed the most optimised builds using Iaijutsu Master completely subvert the prestige class since they tend to be focused around thrower characters ... because the prestige class allows most of its abilities to function on 'the first attack' and giving the Iaijutsu Master Weapon Finesse with a katana ... resulting in builds where you're hurling dozens of katanas at people.

Either way, the main issue with Iaijutsu is the fact it's highly conditional to activate in 3rd ed and that you can do superior damage with other melee or magic builds. Maybe something to think about when creating something for 5e, I don't know. Just a suggestion to go and see how 3rd ed did it understanding that it's not a worldbeater in game terms.