Neither of my favorite builds for Lucian are covered on this site. I thought I would share them with you and I hope you like them.

The first build is Duelist/Split push Lucian. By building BORK and Triforce you are able to proc BORKs passive with Lucians double tap over and over again. This lets you shred anyone who hasn't stacked armor and heals you for a ludicrous amount. These two items hit the sweet spot for AD, attack speed, and CDR. Making Lucian's combos feel very smooth to pull off. These items also make Lucian very fast and allow his E to off cooldown constantly. Making Lucian extremely hard to chase or fight.

The Second build is Front line Teamfight Lucian. With this build you are actually safe to play in the front line. Your damage and mobility are so high that you can poke their front line to death while staying relatively safe. When you see an opportunity, take it! Melt whoever is out of position then pull back to your team. If they try to dive you simply hide behind your tanks and kite away. Avoid split pushing with this build as you are not as strong in a 1v1 as Duelist lucian. Because you don't have sustain or tankiness you will die to the first person with burst damage and CC to walk in your general direction. You are safest when you have some tanks be dash behind.

Items, runes, and masteries are explained in detail under their respective builds.