Armor with modifiers can be unlocked by simply completing any of the story missions, just make sure you open the end of mission chest and pick up the items that appear before boarding the helicopter to end the mission. When back in the main menu choose your inventory tab and the armour option now equip any of the armour pieces you have to unlock your Trophy.
Weekly challenges appear as a random story mission with game altering modifiers that could see the enemy have more health or see you do less damage for example so they are a bit more of a challenge than a normal mission. To start a weekly challenge you need to have unlocked world level 2 where the challenge appears in red on the main world map so select this and complete it, once complete, it will turn green then wait for a new week to start and repeat the process another two times, where at the completion of the third weekly challenge, your Trophy will unlock.