If you want to play a challenge, just select it from the  Challenge dropdown in the Games & Emulation Main Screen. From now on, only the games associated to the challenge will be visible.
If your user in Multi User Mode is bound to a challenge, you can't change the selection in the  Challenge dropdown.
A challenge may consist of more than one tier. If all achievements of a tier are accomplished, the next tier will be unlocked.
With the  show completed checkbox you can control, if you only see the games of the current tier or if you also want to see the games of the previous tiers.
Important: make sure, you don't have any other active filters (e.g. Platform, Search term etc.) or you risk that a game of the challenge/tier is not visible.
Successfully unlocking one or more achievements will be announced (Metropolis Launcher checks for unlocked achievements when a game is selected or after a game has been launched):
Important: if a user wants to unlock runtime-based achievements, Statistics must be enabled in the Settings page.
Also, you get a notification if you successfully mastered a tier:
Of course, Metropolis Launcher will also tell you, when you completed a challenge in all its tiers: