You will need to find every artifact in the depository, then click the depository on the main menu and view them. These are as follows:
Clover
The Map
Powder
Totemus
Heart Amulet
Guitar Hero
Magic Mushroom
Economicon
Magic Ring
The Blood Tooth
The Candle
Dice
Hourglass
Boomerang
Holy Grail
Hermes Boots
Bottle of Death
Holy Shield
Mjolnir
Gamepad
Kunai
Lager
Glasses of Dwarf
Alchemicon
Medallion
FisherMan
Coupon
The Elder Helmet
Dynamite
Diamond Pickaxe
Crying Baby
Helmet of Miner
The Golden Hearth
The Golden Tooth
The Blue Tooth
Ring of Ci8
Lockpick
Cross
Candlevania
Pact
Pact 2.0
Spurs
Cloud in a jar
Emerald
Bob
The rogue is the best character for obtaining artifacts after you have collected the Lockpick. The Lockpick gives you a chance to not consume a key upon opening a chest or door but as the rogue starts with it, you will not unlock it in the depository, nor see it during a run. The rogue's special ability also allows you to pass through dangerous areas without taking damage.
If you're struggling to acquire the last few, search every room to try and make deals with the cat, save keys for locked doors in the hope they're an artifact pedestal room, and try to keep at least 900g on you in case one appears in the shop. Red chests will also always contain an artifact.
Finally, if you see the Sahasrara rune, you can use it on any floor other than 4-2 to get a boss fight and thus a guaranteed artifact, if you win. Using it after collecting the artifact on any normal boss (floors 1-2, 2-2 and 3-2) will allow you to fight the same boss again. If you use it on any x-1 floor, you'll fight the third boss.