As of 20/07/2020 Campaigns are playable as part of the 5.0 update.
They run for 10 weeks each. You do not need to be part of a clan to participate. You do not need to pay to participate.
You will need a total of 100 stars from any number of tasks within the same 10 week campaign period in order to level your artifact to level 1. (You'll need 1,000 stars to hit level 10 for the other achievement) You can earn a maximum of 100 stars each week, unless you have the campaign pass that grants you extra stars each week.
To find tasks, see below - there are 16 available each week.
On the main map page, press Y to pull up the menu and navigate to the bottom left corner, select campaigns.
Press
to change tabs to tasks and pick one to work on, they can range from things like:
Play 5 matches with a specific banner/weapon/class (easy)
Craft items in the soulforge (medium)
Kill X enemies of X colour (longer, but still easy)
and a variety of other options, that I won't bother listing out here, complete any task to complete this achievement.
You can also use gems to "skip" tasks, and this counts as completing them also, but they're so easy it's not worth the relatively high gem costs to do so. But it would save you time and you still get the stars.
Note antyagonista points out that you may have to collect your rewards from the tab in order to unlock the achievement, I was rushing so can't remember what I did. But why wouldn't you correct your rewards after doing all those tasks
Please let me know if you need any further help.