With a nerf to FN-2199 - Loyal Trooper believed to be looming, consider this a head start on the Kylo/Phasma build that will likely follow it.
This is pretty aggro-leaning. I've made choices that push toward damage instead of control, and while some cards are dual purpose (I'm looking at you Ancient Lightsaber and Clash) most are aimed at doing more damage and/or turning your dice to damage. Kylo has 3 melee sides on each die so the modify sides on lightsabers usually work out. It also means Lure of Power produces damage quite reliably, though its interactions with and can be better in some situations. I have kept an eye toward planning for one character's demise, so most upgrades can be used on either of them and several also bring redeploy to the table.
Building on a Budget
If you can't afford some of the pricier things here, there are some easy swaps you can make. Ancient Lightsaber can be replaced by Praetorian Guard for a bit less damage, but the useful guardian ability. Similarly Z6 Riot Control Baton can easily be replaced by Lightsaber from the starter set for a small loss of damage consistency.
Secret Facility - Scarif - This is a bit of tech to help speed up our mixed damage. A lot of other decks can't make good use of it; Thrawn/Unkar won't have damage sides, and decks with predominantly or can already resolve their damage with only 1 action. So it can help alleviate our need to take 2 resolve damage actions by claiming instead, while also offering little to many opponents. The other main contender here is Obi-Wan's Hut - Tatooine which is good for Kylo, but blue characters are fairly widespread so many other decks can also use it well.
Ancient Lightsaber - This die is about average for a 2-cost upgrade. It has a +3 side with good synergy with Clash. It can also be trashed to heal the character for 2, which should by good for Kylo who will likely be focused.
Crossguard Lightsaber - Kylo's favorite. The die is average with 5 or 6 (vs shields) damage on it. The 1 damage on play is what makes it worth having over other similar choices.
Force Illusion - Solid damage mitigation. One of only a few defensive cards. Because there isn't much defense here, the picks are all high-value ones able to stop a lot of damage at once and/or highly versatile.
Lure of Power - Perhaps a controversial pick. It's here because it can do more than just damage. The total +4 here is a bit underwhelming, but it makes up for it in being able to go on either character and being able to modify either damage type and various other symbols as well. Especially dangerous when adding to , and in early game + plays are huge.
Z6 Riot Control Baton - It's a monster. Overwrite it (or any redeploy upgrade) onto the character that's focused so you don't lose the value of their non-redeploy upgrades. This is just as good on either character because all the damage is unmodified, and Z6's power makes it one of the most consistent damage producers in the game.
Clash, Force Strike, Lightsaber Throw - A whole kit of cards that do damage. Force Strike doesn't do much if Kylo is dead. Clash is more versatile in that it can turn Z6 too, and likewise can wreck an opposing melee die as well. Lightsaber Throw is probably most reliable of the 3; it's quite easy to show melee in a variety of ways. Feint competes with these for slots, and I feel the cards here are more reliable than triggering Kylo's guessing game for maybe 2 damage Since it's two-ish damage for all of them, I'd rather have the other restrictions, especially since Feint is utterly dead if Kylo is already toast.
Doubt, Isolation, Rend, The Best Defense... - Not everything can do damage. These are some of the best tools available to mitigate opposing cards/dice. Rend might seem odd at first, but helps a ton against Imperial Inspection and also hinders some decks that rely on the Sith Holocron engine. It's so good against a few heavily-played cards it feels like a must-have. Note there is no Deflect here; I avoided any narrow choices, opting to make the build more broadly useful instead. If your meta supports Deflect then swap it in, but I caution that against some decks (like Thrawn/Unkar) it's just terrible.
Enrage, Logistics - With a bunch of 2- and 3-cost upgrades and Kylo's pay sides, this deck is fairly resource-hungry. These help alleviate that. These are among the top things you want to mulligan for, having one or two extra resources is huge.
Things That Didn't Make the Cut
Vibroknife - I feel everything here is better. Lure of Power is probably closest to losing its spot to Vibroknife, but honestly the Vibro power is rarely relevant in a deck with such a huge variety of damage. The potential damage sides are the same, 1/1/2, and I prefer the flexibility of Lure modifying anything and working well on either character. Vibro does have the benefit of printed damage sides which Facility can resolve, and works with Clash and Lightsaber Throw, so it might end up winning out if consistency in using those cards is a bigger problem than I currently think it is.
Makashi Training - Damage is basically the same as other 2 costs upgrades, the power is great... but it spots blue and the power is irrelevant in some matches. I'd avoid unless you like to live dangerously and also play in a meta full of melee.
Leadership - A couple considerations kept this out. First, 2 dice of Phasma are really productive already. You trade away a lot just to ready Kylo. Second, it's a dead draw if either character is dead. Third, the cost is tough too, as are any rerolls you would need to do on Kylo's second activation. Ultimately just too situational; losing a 1-die activation to ready up papa Vader in a deck with Boundless Ambition works well, but here we're losing a lot more and gaining a lot less, so it's difficult to justify.
Tactical Mastery - Another card where the price is a major factor. Going fast is great, but we don't really have any tricks that need it, so it's not a very valuable play to this deck.
Force Speed - I'm way more on board with this as a source of double actions. I've left it out for now as I don't see anything clear to replace to fit it in. Possibly slot it instead of Rend if you don't fear Imperial Inspection or Sith Holocron, or a compromise position is running it instead of 1x Doubt and 1x Rend. I consider it completely non-essential for this deck, but if you have them or don't mind picking them up go ahead and run it if you like.
Feint / Hate - I talked about this a bit earlier, but the gist is Feint can do 2 damage, just less reliably than all the things I did include. It's better if you are playing Hate in your build, but I don't see the point of spending 1 to have Kylo punch himself; he is already likely to be the focus, and putting Hate on him would be the only way to make Feint decent value. I might ultimately change my mind on this, but it depends just how much mill is in the field, it's one of the only decks where Hate presents a clear advantage since they aren't trying to win by damage. If it takes over everything, I would make room for the Hate/Feint setup for sure.