How do I get artifacts in Risk of Rain 2?

You need to find a code, use it to unlock a challenge in a hidden part of Sky Meadow (stage 5) and then complete it:

Unlocking

To unlock Artifacts you need to use the correct Artifact Code in Sky Meadow, using the Compound Generator below the map, then interacting with the laptop. A Portal to the trial for that Artifact in Bulwark's Ambry will spawn.

The Compound Generator will cycle in the order: square, circle, triangle, diamond (■ ● ▲ ♦)

In Bulwark's Ambry, the Artifact whose trial is being attempted will be active and an Artifact Key will be on an elevated block in front of the Artifact Reliquary. The player must give the Artifact Key to the Artifact Reliquary, which will cause monsters to start spawning, with more Artifact Keys dropping from randomly killed monsters. Using an Artifact Key will remove them from all player inventories.

Upon destroying the Artifact Reliquary, the Artifact can be taken and a portal to a First Stage will appear.

I think so far every stage or environment (including hidden ones) have exactly 1 hidden artifact challenge code. You should search for codes on tablets like this one:

If you want to find all artifacts by yourself, I recommend this Artifacts Hints page: it has some vague hints about stages and where to search for the code.

If you just want to unlock them without searching, here are all artifact codes under this spoiler:

Sources: source 1, source 2 .