Ok, so if you're like me, you finished the story, finished the survivals, altars, everything else that disappears after you finish, and Mayhem doesn't help much! Sadly, there is no way to do this without some serious grinding. But hopefully I found a way that makes it much easier!
The best way I found to get a fair amount of these challenges done is to go to the Downtown area (bottom left island) and cause some chaos there. I've tried the other areas but Downtown gave me more demons attacking me at the same time. This helped a lot for one of the most annoying challenges, Energy Caster Arcs. Thankfully in the Downtown area they mostly stick together, even the flying demons, so I finished it in a matter of minutes. Same for Damned Impaler ultimate upgrade challenge, kills with the weapons, catching the Sin Orbs, igniting enemies with the lava cannon, so on and so forth. It was just easier in Downtown because the mass of demons sticking together and there's a good number of them.
Another annoying challenge is upgrading all 20 weapons (not including the pre-order DLC weapon). Again, be in Downtown because of all the demons, just use Diamond Sting (7 Deadly Weapons - Greed), kill every demon you see with it, then use the vacuum stomp to suck up all the wages they drop. Go to a little vending machine shop, buy the appropriate ultimates for weapons, maybe damage and accuracy upgrades for weapons you do not need ultimate kills with, and repeat.
The grind is very real for this achievement, but it is the fastest I have done so far. I was at 60 challenges completed earlier today (1/29/15), now I am up to 91.
Hope this helps everyone else. Good luck!