Predecessor Builds > Serath > Item explanation at the bottom

See bottom for explanation
We start Crimson Edge into Infernum, because Skyspliter only gives you 3% more lifelink. Skysplitter's passive is on-hit, while Infernum's is not. This means Serath's primary applies full infernum stacks for 100% of Infernum's passive damage, versus 50% damage from Skysplitter. Additionally, enemies don't have enough life early for Skysplitter to make a huge difference. It's value is largely in its attack speed, and it's lifegain until mid-to-late game.

After Infernum, get Ruthless Broadsword, because it has most of Basilisk's passive (armour shred). Then you either fast track Fire Blossom, or build tainted blade, depending on how easy/difficult the opponents are.

Once Tainted Blade is done, go back to finish basilisk.

Earthshaker will be your last item, although you may want to finish Skysplitter first.

Item order explanation
Infernum is your single largest damage source from items, followed by Fire Blossom. The rest of the build just makes you hit harder.

Ruthless Broadsword comes before tainted because it has a lower gold investment to get the initial passive. We then finish Tainted (after Fire Blossom) first, because it increases damage received from all sources (Infernum, Fire Blossom, Ultimate's true damage DoT) by 8%.
Basilisk only increases physical damage by shredding physical armour (rip magic/true damage).

Earthshaker comes last because while it does increase ability damage by up to 15%, it does not increase the damage from items, or auto's. Additionally, it benefits only you, while Tainted Blade increases the damage your allies wil deal in team fights/ganks.