What do I need to do to earn the tier 6 mission chest reward?

It's all about Total-Score. The earlier the mission (or the earlier in Story-Mode), the harder it is to get a higher Total-Score. So even if you max (All Plat) Utility/Building/Combat + All Bonuses + All Mission Side-Quests (Drone Landing, Storm Chest, ect..) The Total-Score still won't reach the minimum required to pop a Level 5/6 chest. The lower missions simply don't have enough potential score methods. A level 28 Mission can yield a level 5 chest pretty easy if you complete a few side quests, or a Drone Landing (Those things give a LOT of Exp). If you and a well organized team push through a 28 Mission you can get a level 6 chest. If you and a half-organized team jump into a level 40+ mission, level 6 chests are not to bad to snag. Though, will that that said, level 6 Chests just give a bit more of what you're already getting in a level 5/4/3 chest. So if the time it takes you to run ONE mission for a level 6 chest is longer then it would have taken to get two level 4 chests, it's kind of a waste (unless it's a Time-Reward for something you want, then it's worth the extra effort.