Since patch 8.4, a new item has been introduced:
Spellbinder

This build is based around getting this item and abusing it's active and roaming potential.
Spellbinder is basically Deathfire Grasp 2.0

Primary path:

Electrocute - Take this because of the raw burst potential.

Sudden Impact - Take this because it simply outperforms the other choices for this build.

Eyeball Collection or Zombie Ward - Eyeball Collection is nice for ensuring that you snowball to dominate the game, Zombie Ward is a viable alternative if you're making use of your wards.

Relentless Hunter - This will help you roam and snowball the game in your favor.

Secondary path:

Manaflow Band or The Ultimate Hat - Manaflow Band is going to allow you to rush your Spellbinder without having many mana issues, if you plan on going with The Ultimate Hat you'll want to at least get the Lost Chapter so you can sustain yourself in lane.

Celerity - Not only does this improve your roaming potential, but it works well with Ahri's Orb of Deception, and the passive AND active movement speed from Spellbinder giving you a massive boost to your ability power momentarily.

Standard Start:
Doran's Ring is great for sustaining yourself in lane and making sure you have a healthy pool of mana to draw from.

Snowball Start:
Dark Seal is only good if you know you can kill your lane opponent and roam for kills.

Core:
Spellbinder - This item is cheap, has good stats, has a crazy passive/active.
Sorcerer's Shoes - Great paired with Spellbinder and the Celerity rune, more damage and magic penetration.
Luden's Tempest - You need some CDR and Mana, and this item's passive pairs perfectly with the rest the build.
Rabadon's Deathcap - More AP, more damage.
Void Staff - This is always a good item to have beyond mid-game, especially if they start getting magic resist.

Situational:
Lich Bane - You already deal quite a lot of damage, but this item gives you everything you want. Consider getting a more defensive item, unless you feel confident you don't need one.

Liandry's Torment - You'll want to get this if you're having trouble with poking/killing their tanks. Once the tanks are dealt with, you're free to assassinate their carries.

Morellonomicon - Get this if you need the gervious wounds to be applied to an enemy with high sustain that you can't burst down reliably.

Banshee's Veil - Get this if the enemy team has game-changing abilities that are hard to react to, and/or they have a high magic damage team.

Zhonya's Hourglass - Get this if you're good at using the active to avoid game-changing abilities, and/or they have a high physical damage team.

Rylai's Crystal Scepter - Get this if they have a bruiser or tank which is somewhat kiteable, you'll want to pair this with Liandry's if you do plan on getting this.

Mercury's Treads - Instead of getting Sorcerer's Shoes all the time, consider getting these for a CC heavy team comp. It's passive may not seem like much, but this could make the difference between you living or dying sometimes.

Oracle Lens - This item is risky, but it can help you stack your Eyeball Collection and you can catch people out face-checking bushes. I don't recommend getting this if you have poor map vision or a lack of awareness of enemy team positioning. Definitely do NOT get this if you're using Zombie Ward.

Misc. Items:
Control Ward - Try to have one on the map at all times, placing it in spots that're fairly easy to defend(your teams half of the map).

Elixir of Sorcery - Try to get this if you're full build or if you're about to push to end the game and you have nothing better to spend your gold on, or you don't have enough space to purchase an item better than it.

Rabadon's Deathcap (Masterwork) - If you have an Ornn on your team, try to make use of his passive and buy this ONLY if you have no more items to build and you have spare gold laying around.

If they're in range of charm:
Charm, if it lands, Spellbinder> Spirit Rush+ Fox-Fire+ Orb of Deception. Use your remaining Spirit Rush charges to escape or ensure the kill, which ever way means you don't die.

If they're not in range of charm:
Spirit Rush and/or Flash+ Charm, if it lands, Spellbinder> Spirit Rush+ Fox-Fire+ Orb of Deception. Use your remaining Spirit Rush charge to ensure that you're safe.

If you miss charm:
Missing charm can have an impact on getting the kill or not, but it is not the main source of your damage. If you know you can still get the kill and get away safely, commit to it.
And try not to be too tilted after missing charm.