To complete a Weekly challenge you must start it from the SMI menu (on the Paladin).
Once you are on your world map hit the Y button and access the SHADOWnet
All your challenges appear at the bottom of the screen. There are weekly, daily, in-game and friend challenges. Move over the one you would like to start and hit the A button.
There will be a short description of what to do for this challenge. Make sure that you have the requirements ready to go before you start the challenge just to make your attempt at it easier. Hit A again to start playing towards that challenge.
Daily challenges restart after 24 hours (1 AM PST)
Weekly after 7 days (Monday 1 AM PST).
Complete the requirements for that week and your achievement will pop in game.
You also might need to have activated the online pass codes for this game. I have activated them, however, some people are having difficulties seeing the achievements.
If you can not see the challenges and have not activated uplay that might be the problem.