Somewhat contrary to what I assume would be everyone's first instinct, I've found the most success simply ignoring a lot of the big enemies and snapping up opportunistic kills, or preying on groups of weaker enemies. Too often the big enemies are surrounded by a lot of smaller ones that love to interrupt your attacks and generally tack on extra damage, keeping you tied up to get hit by the big one's stronger moves. I suppose this is what the powers are for, although I have absolutely no idea how much damage any of them do. (Or how much HP the enemies have, for that matter.) I imagine that different enemies will be easier/harder for specific characters, so that sort of knowledge will play a big part in efficiently getting things done.
According to the ingame tips I've seen, elemental weaknesses and resistances are in effect for some enemies. Polar Bears, for instance, are allegedly weak to fire and flinch away from it when normally they have constant super armor. And some enemies have weak spots, although it's tricky matching names to enemies without a picture guide.
In general the mode gets easier as you rack up the boosts, so it's probably best to use your powers as you need early on instead of saving them for later. That said, it's probably a good idea to keep health recovery available as a last resort.
Lastly, there does seem to be a fair amount of luck involved regarding where you spawn, what enemies decided to harass you, and what sort of challenges you get behind the doors. It's easy to just have a run of bad luck and fail to amass any boosts of note.