Like other said a volcano would darken the atmosphere.
The enemy of my enemy is my friend so maybe our intrepid hero gathers together a team/gang/club/"do gooders are us" to help him out. No matter how evil you are you will still hate Lex even more.
However, our lex luthor has a plan in mind, for world domination. He can't total wreck his world because he lives on it. He does have bunkers of course as a last resort.
He turn the world dark to black mail the residents of his world for full control. So Lex brings in trucks of special compound X, and triggers super volcano. Compound X is a nano cloud that feeds on the ash, but also alters it so only the shades of red and blue for growing can pass through, and infrared for heat.
Your world won't be pitch black, but with only red and blue passing through,it definitely going to be dark and eerie.
So the stage is set, and the whole world looks like below with infrared for heat and maybe some other color/spectrum. Lex is blackmailing the world, and its nearly hero time.
Now Lex is knows he is in for a long siege, and residents are not going to give in that easily. There is still backing stabbing on all sides. He will be attacked on all sides, but also has a plan for undoing it. A nearby river is prepared and safe guarded to dump on the volcano to extinguish it. Also the nano cloud, could be used to clean the sky in a few months. However, Lex will never do this for ages to ensure his dominance.
Enter our intrepid hero. Shooter guy. haha
Some of the Lex's forces he can sneak by, some he's going to have fight. After a long trek to the Evil lair the hero and his intrepid band of followers (if needed) comes to the control room. Lex, of course is waiting inside, and every one else fights his minions.
Lex, leads our hero along narrow passages,needless sharp cliffs, and other traps where he has prepared for the final confrontation. Either that, or some stupid complex torture chamber. Lets MacGyver that. Lex and hero guy end up using swords because Lex wants to kill him slowly. Through some surprise or whatever the hero ends up winning.(he stumbles, his Mom's name is Martha, hero talks him down, or any other movie trope.)
If desired you can force intrepid hero to hack Lex's computer, and any other shenanigans you wish
Alternative he can just push the "big red button".
The flood gates open, extinguishing the volcano. The nano clouds are re programmed,and start purifying the air and allowing light back into the world.
The hero has to rally a bunch of people who really hate Lex to divert a river, or cause an avalanche of snow, or some other way of putting it out.
Additionally, since the nano clouds are swarm controlled you can force the hero into all sorts of different hacking,bribing situations to your hearts content to break the encryption, and steal control of the swarm and purify the air.
Maybe the whole sky is one big nano cloud, and there's no volcano at all.
Maybe Lex disguised the nano cloud as pollution from Lex Corp's factories until it was too dense for any one to get rid of.
Space balls the password is "12345" no "Lex12345" hahahaha, maybe not "!Evil!Bad!Guy!12345:(muhahahaha" more secure
I have read darkness increases crime as its easier for criminals to operate unseen.
It isn't just cloudy/... all the time.The world is supposed to mirror ours. cloud cover and all. $\endgroup$