During the Season of the Hunt in Destiny 2, your Seasonal Artifact will be the Fang of Xivu Arath. Getting the Seasonal Artifact is actually quite easy this season. First, install Beyond Light and then load up the game.
Now, play through the introduction mission and then head back to the Tower after you meet Variks. There is a new contraption in the tower that you can find if you head to the walkway between the Gunsmith and the Annex area. Here, you will find a red elevator.
This will bring you down to the bottom level, then turn left and follow the corridor to a new room. In the corner is a large view screen, and you can get the Seasonal Artifact by interacting with it.
Just like previous Seasons, the artifact can be leveled up by earning experience just by playing the game. As you earn points you will be able to purchase new mods that will help you to deal with difficult Champion enemies, and that will give you unique benefits.
You only have a limited number of points to spend, so use them wisely. You can, if you wish, reset your Artifact to get different mods, however, so keep that in mind. You can earn XP by playing Gambit and Crucible matches, Strikes and Nightfalls, or grinding bounties, that most glorious of Destiny 2 traditions.