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.
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.
Please let me know if you need any further help.