So firstly, this is what I like to go, gives you some early damage and mid game power and utility. The borc cleaver build i feel like is more well rounded in comparison to the other builds like borc tri, full lethality, crit. They each have their own perks but this is a personal preference.

Borc cleaver's primary bonus is the early damage and the cleaver procs which help your ult do a lot more damage.

Borc Tri was the "old" build that works completely off of Lucian's passive, but you were kind of forced into biulding cleaver on this anyways to help augment the ult damage.

Full lethality is rather useless because you're essentially a mf/jhin/ashe, an ult bot. Every team fight you basiclaly press r and do damage that way because your autos won't do much damage.

If you get realy fed early, it's justifiable to go the crit build. First back into BF/caufield and just punish the enemy with massive amounts of early ad. You have no problems with cdr nor mana because you're itemizing essence reaver. But with this build you have no armor pen/lifesteal until 4th item 5th item because of how many items you need to build to get decent %crit.

Standard runes and masteries:
ad reds, mr blues, armor yellows, attack speed quints
masteries I'm on and off about taking natural talent instead of the lifesteal/spell vamp
i prefer the lifesteal

So if you bothered to lookat my boots section you see I put three options: tabi zerk and merc. Choose based on what you're up against although personally I do not go zerk often.

With the current popularity of lethality, buying tabi after you buy the main components of borc (recurve bow + cutlass) or after the completion of borc makes you super durable against them with the extra armor.
Even buying tabi early in lane against the enemy bot is super helpful if they are super dependent on auto-ing you for damage, good examples of this are like kog/ashe/sivir etc.

Merc Treads are obviously for ap heavy comps with cc that you want to get out of quicker.

Zerk greaves are my least favored choice because we're already running borc+pd which gives you decent attack speed. The reason for not alwasy defaulting for this is because lucian doesn't really rely on autos to deal the majority of his damage.
His damage lies in the combos with the double tap passive and with the buffs to the fluidity of his q/w, you can auto-cancel a lot faster so there is no real need to be itemizing heavily into attack speed.

If you're not a fan of the 20% cdr and you also want the reduced cdr on flash/heal, feel free to go lucidity.

So core build is: borc + cleaver + pd + IE + boots
The ie is there mainly to give you a big boost to your ad damage and to complement the %crit you get from pd. last item is very very situational and up to you.

Amongst the items listed, mercurial scim and the last whisper variations are the much preferred but obviously have their uses.
Merc for when there's one specific cc you need to avoid Ex: malz ult, ashe arrow etc

Last whisper for when there is at least one super beefy tank (aside from the supp) that you need to cut down, obvoiusly itemize which one based one the tank.
Go Mortal reminder for the obvious choices of vlad/ww/illaoi/maokai etc that have built in heals in their kits. otherwise go for lord dominiks when the enemy tank is like naut/poppy

If you do not need to be the primary damage threat, sometimes going GA is great for the utility so there's less incentive to be focused by the enemy team and you can just sit in the back line and do damage.

And if the enemy team is also prioritizing that lethality, stacking an iceborn on top of the tabi that you should be building would be very helpful in making you tankier. Of course iceborn would augment your damage with the sheen procs and help in the chase, as well as giving you the extra cdr.

Gunblade is personaly a troll item but it does help your w do more damage, you can slow the enemy with the double actives (borc + gunblade) and gives you more healing with the unique passive.
Bloodthirster is obviously the more heavy lifesteal choice.