I used to go botrk into black cleaver most of my games, but lately i find BotRK to be a waste, teams are not stacking health/resistances, everyone are squishy trying to 1 shot each other.
Crit : Gives you better burst for squishies (obviously) better lategame transition aswell, and snowballs really hard, your passive makes you crit a lot more even with just ER since you always double tap after every ability.
Botrk: If you are gonna kite a lot, Juggernauts, tanks, Divers - takes Botrk into black cleaver, gives you a lot of CDR and the botrk active is almost like an extra life line against Juggernauts and tanks who get ontop of you, the life steal also helps you duel with low damage targets in long trades, black cleaver helps shred the armor not only for you but for your entire team aswell, and you shred it real fast thanks to your passive.
Not a Lucian main, but i play him a bit, so take it with a grain of salt.