Simple maths (in level 18):
-Damage per shot:
65 + 40%AP + 33%AD
In this build at least 145 damage per shot
-Number of shots:
7.5 + 10.5*AS
In this build you'll shoot 28-29 shots
-Total damage = 4060 to 4930

Runaan's Hurricane:
-I haven't seen this item in much builds, but it's worth it. With Lucian, after every ability you shot two bolts (the second one deals 75% of the AD). With this item you shoot two bolts always and after ability you shoot four (two that deal 75%).

Blade fof the Ruined King & Bloodthirster:
-Just more AD, AS and an enormous amount of life steal which makes much easy to survive team fights and getting buffs, regenerating life in a Lane Swap,...

Trinity Force:
More AD, AS, and a minimum of AP to deal a great amount of damage with the Ultimate.

Last Whisper:
-To face Armor champs like Rammus, Khazix,... (Never 1v1!)

Play safe. With this champ it is really easy to get outplayed so avoid any poking from enemy bot lane, if jungle appears immediately under turret, never dive if enemy is not in three hit kill health or less,...
Farm with precision, minimum 10 more farm than other team's ADC at minute 20 (At least 90 farm).