Hunter's Blessing

You're first going to want to get Hunter's Blessing and tier 1 of Devourer's Gauntlets. Hunter's blessing is a must for Artemis in the early game because your main goal of playing the adc in duo lane is outfarm the enemy adc and one of the ways you can do that is getting last hits on the minions to get that bonus gold. Hunter's blessing will reward you for getting those last hits and when it is fully stacked you'll be able to see a notable difference in your basic attack damage vs when you didn't have it fully stacked.

Devourer's Gauntlet

The next item you're going to want to complete is Devourer's Gauntlet to start stacking last hits for that item as well because more than likely when you have enough gold to buy tier 3 Devo's you will still be collecting stacks for Hunter's Blessing as well. Devo's will be able provide with much needed sustain and damage once it's online.

Warrior Tabi

Personally I prefer Warrior Tabi over Ninja Tabi because of the extra power you receive which makes poking the enemy duo easier with her 3. It also powers up 4 as well to help deal damage when using it. Once Artemis' 2 is leveled enough it makes up for the lack of attack speed that Warrior Tabi doesn't give anyways.

Odysseus' Bow

This item is SO good for Artemis because it not only increases her attack speed to keep up with other hunters without her steroid but the passive helps her wave clear so much easier and safer since you don't have to use 3 on the wave and can use it poke the enemy adc instead. Around the time when have enough gold to buy O-Bow is when Artemis' 3 does a really good chunk of damage at this point in the game especially if you've been fed a kill or two.

The Exceutioner

Executioner is a definite pick up because of the stats it gives and is cheaper than Qin's Sais (2350g vs 2700). Since Artemis is mainly a basic attack hunter instead of the mage-like hunters like Ullr, Excecutioner is actually better for Artemis to help increase her attack speed and to also help shred against protections because at this point the enemy team most likely has defensive items already built and Executioner will help rip through those protections.

Titan's Bane

By the time you have gotten Titan's Bane it will be sometime around late game depending on how the match is going. Titan's Bane is simply here to help even further to shred through protections because at this point the solo lanes should be joining the fight with their natural protections and their defensive items so buying Titan's Bane will allow you to keep doing as much damage as possible to them.

Qin's Sais or Deathbringer

Depending on the enemy comp you will either want to go Qin's Sais or Deathbringer.

Qin's Sais is normally what in my ranked games because late game you're going to have to worry about the supports and solos having massive health pools and Qin's Sais will help whittle them down faster than what Deathbringer can do.

Deathbringer is chosen over Qin's Sais in some cases when the enemy team is mainly composed of squishy gods like a mage support or mage solo. The damage increase from a random crit will benefit you a lot more against squishier teams than Qin's Sais but if the enemy team is fairly tanky you're going to want to take Qin's Sais.