Here it is! The best one!
I tried three separate games / ideas and none worked. Will get into more detail as I try again, though I will say it's not just "have AZ be the last man standing" or "win with at least one copy of each Ongoing in play", in case folks were going that route.
My initial attempt was unsuccessful: I played at least 1 of each of AZ’s Ongoing cards, including both Impales and 2 Sub-Zero Atmospheres at once. I finished him off by using Fueled Freeze to destroy Impale, Cold Snap, and Sub-Zero Atmosphere. I also played and used the powers on all of the other Ongoing cards.
I feel like I’m on the right track, but just not quite getting it right.
Maybe you can’t play Modules? I don’t know why you wouldn’t, since FF5 AZ isn’t missing equipment like Termi-Nation is.
Maybe you shouldn’t destroy them; maybe you have to have at least one of each at the end of the game?
Maybe you need as many of them in play at once as possible. Difficult, given AZ has normal card play speed, and only a bit above average card draw, and no search for anything but modules.
Got it: https://pastebin.com/caahsFgB
My guess was that it would involve dealing self a bunch of fire damage and having at least one of each ongoing in play, so I did that. I also dealt Progeny a bunch of cold while Scion of Frost was around (just because Frost was easiest to deal with and some ongoings forced villain damage :p).
How much fire damage did you do? I did a reasonable amount (13 in one turn once) but didn’t get it.
Also, did you have them all out at the end of the game? (Because I didn’t do that either)
No idea how much fire damage I did, but I'll dive into my log and find out (though I think that 13 was also the most that I did in a single turn). I did have one of each ongoing out at the end, but was too impatient to get all possible copies of the non-limited ones.
Excellent. I love unlocks like these, I can give it my best shot and not have to worry about it for more than a couple hours. Don't forget to claim your hall of fame spot!
Analysis: a0 fire - Pastebin.com
A0 was dealt a total of 42 fire damage across 11 different turns (2+2 from Progeny, the rest from self). The most in one turn was 13, the least in one turn was 1.
How do I do that? 
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Shot-in-the-dark guess:
Complete the pre-requisite games. Beat Progeny in Megalopolis with the base Freedom Five. At the end of the game, Absolute Zero must have one of each ongoing in play and cannot be incapacitated. Absolute Zero must have been dealt Fire damage on at least 10 separate turns.
Alternatively it may be 10 damage at once, since if the ongoings need to be around until the end then it would explain McBehrer's failed try.
Or it’s at least 25 or so total, dealt by himself.
Important: did you get it when you won? Or when AZ dealt himself the last instance of damage?
If it was even you won, who got the final blow? I’m really close to the end of my game, and I need to know.
I got it when I won, and A0 did the final damage (it was the reaction cold damage from self-fire from Frost Bound Drain)
Got it also. Had one of each ongoing in play at the end, AZ dealt the final blow. Also dealt 10+ fire damage to himself in at least 1 turn. Unlock log is here: https://pastebin.com/pJhuFp03 Good luck everyone!
Got it. Had one of each ongoing and equipment out. Did a LOT of self fire damage, but kept at full. Won with the reaction damage from Cryo Chamber, so even though I used every power, it didn’t leave play.
This seems like a pretty straightforward one. We just need to figure out what the specifics are.
I just got it. I don't think duplicates are a thing. You can probably get it only having one of each out. The highest damage I dealt myself was 14. Progeny went down to AZ's start of turn damage from his ongoings. Here's the log.
Got it, too, on my first try, but my log is incomplete–I actually would have gotten the hall of fame spot if I’d been able to finish my game right away. I played all ongoings and equipment but didn’t use the power on Glacial Structure. My biggest hit was 27 fire->30 cold to Progeny for the win, but I’m sure -that- isn’t necessary.
You don't need to use all the powers to get the card. In my run, the only power other than the base power that I used was "thermal shockwave."
Got it. Put all 5 ongoings in play, both modules and the cryo chamber.
Most fire damage Zero ever dealt himself was 11.
Didn't count the number of turns or amount of fire damage or healing yet.
Used powers on Glacial Structure and Thermal Shockwave.
Only One-shot I used was Onboard Module Installation.
Here's the log.
edit: Zero took fire damage 10 times for a total of 42 damage. Zero received 53 points of healing in the game.
theory
Prereq games, vanilla F5 in Megalopolis v. Progeny
win the game. All 5 of Zero's ongoings must be in play. The Cryo Chamber must have entered play. Zero must take 10 fire damage in a single turn. Zero must heal 20 health over the course of the game.
No unlock in my mobile attempt.
Not sure what I did differently than those who unlocked, so either I forgot something or we’re missing an element.