All of the information in this guide is correct as of Season 15, Season of the Lost. I have finished my time with Destiny 2. If there are any changes to this achievement in a future update it will be up to someone else to write the guide.
With the Forsaken update what was considered "Challenges" were changed to "Bounties" and no longer work for this achievement.
Challenges have changed once again with the Beyond Light update. I logged in and my character was at 1100 light level.
There are several weekly challenges that you can complete each week. There are weekly challenges for doing bounties for NPCs in the Tower. There are also weekly challenges for doing Crucible, Gambit, Strikes and Nightfall Strikes, and dungeons. You will know there is a weekly challenge when there is a gold star by the option on the Destinations tab.
When you highlight the NPC on the Tower map in the directory it will show the gold "weekly challenge" under their description. Go to that NPC and you will be able to purchase 4 bounties that have specific requirements with your glimmer. The bounties only last one day after purchasing them.
The NPCs I was able to find in the Tower were
Zavala (strike bounties)
Banshee-44 (gun bounties for various weapons
Lord Shaxx (crucible bounties)
The Drifter (Gambit bounties)
Suraya Hawthorne (clan bounties - mainly for raids)
You can purchase a 5th bounty from the NPCs for more glimmer than the normal ones that gives a random requirement.
Just doing what the bounties requires does not automatically complete it. To claim the bounty reward you just open the directory and go to the Quests tab. From there all the bounties you have aquiried will be on the right side of the screen. The ones you have completed will be at the top. Click on them to claim the rewards and have them count for the weekly challenge.
For the playlist items you only need to do however many matches it requires. Generally Gambit and Crucible want 3 matches each. Nightfall Strikes have percentage bar, where harder Nightfall Strikes fill the bar faster. Once you have completed the required number of matches the weekly challenge automatically completes.