Hello,
im a lucian main since season 3 and loved this champ since.
After a lot of testing those builds are the best for Lucian.
Lucian is a close combat marksman, having a strong 1v1 but also a pretty good teamfighter.

This is my favourite build. I´ve won countless matches with this build.
Bloodthirster first gives you the sustain you need and the 80 AD will help you deal the ULTIMATE DMG.
Manamune second gives you a nice mana pool. Lucians Mana is quite low on its own, so having a bit more is very nice.
The boots of swiftness is strong choice, as many of the META champs have a lot of mobility. The movementspeed will help you dodging a lot of skillshots and repositioning yourself.
Youmus is a good choice for the 3rd item as it gives you movementspeed and a lot of burst which will come in quite handy in later teamfights.
Seraphs embrace gives you a nice shield with its active ability and the mana can save your life in later teamfights.
The last item will be runaans as a strong teamfight item for the AOE DMG in those baron teamfights.