You seem to be looking at not just a city based level of pollution, but a planet wide level of pollution - where the entire atmosphere is polluted so much that it blocks out much of the sunlight even without typical clouds.
So building skyscrapers really tall is a way around this.
To build skyscrapers you will need a solid foundation - they can't be built just anywhere, they need to be built on a solid layer of bedrock and ideally far from tectonic plate boundaries.
Definitely check out the geology that helps support the skyscrapers in New York.
In terms of how high these skyscrapers need to be to get the effect you want (judging by the screenshot you included), this depends on exactly how far up the polluted atmosphere reaches. There is a speculated maximum height for skyscrapers, but if you start going too tall you will run into problems with gravity and the spin of the planet, not to mention structural problems that might require you to invent some new wonder material to make it all feasible (eg the fictional material of adamantium). Carbon nanotubes might work, but currently we can't build a structure bigger than 1 metre with carbon nanotubes.
Something the size of the Burj Khalifa might do the trick - which stands 830 metres tall or 163 floors high.
As you can see there are several other normal sized skyscrapers which are just dwarved by the Burj Khalifa. I am imagining a level of pollution that covers these smaller buildings, but the top half of the Burj Khalifa is poking out of the clouds, something like this (an actual photo of the Burj Khalifa while it was still under construction):
The thing that concerns me is the pollution.
You will need a delicate balance - too little pollution and you won't get what you are after but too much and the surface will be virtually uninhabitable. Too much pollution could end up making the whole planet incapable of sustaining the lives of the people living in the skyscrapers. In other words, so much pollution that there would be limited plant life and limited sea life - and this is where most of our oxygen comes from. This could easily be a dead world with that level of pollution.
You could have skyscrapers dedicated to farming to overcome this. If this appeals to you definitely have a look into vertical farming. But you would probably need half a dozen or more farming skyscrapers to support a single habitation skyscraper.
If this planet is part of a star travelling empire you could have orbital farming platforms in space or have regular deliveries of food from an agrarian planet. Having said this though, if you can have orbital farm platforms you can have orbital habitation platforms too - I don't see as removing the need for these huge skyscrapers, however you might want to rethink your social dynamic as the orbital platforms might be where the rich live and the poor live in the skyscrapers of the dead world below (and perhaps primitive wasteland tribes struggling to live on the surface outside the skyscrapers).
Polluting the planet this much would have severe consequences. But you still might get the effect you want with some tweaks.
First consider the idea that this polluted sky isn't permanent all year round - you could have a "smoggy" season where things are this bad but for the other three-quarters of the year the air is breathable and sunlight can reach the ground without issue. Still enough of a reason to build massive skyscrapers to get away from the smoggy season, but not so bad that it kills off all plant life on the surface of your world.
You could also use something other than pollution to get the effect you wanted. A huge series of dust storms that wrap around the planet might give you the "skyscrapers popping out of clouds" effect you want but without pollution. Plants would still suffer, but outside of the diminished light that reaches the surface, dust storms would be less damaging than actual pollution, which is usually toxic and harmful to all life.
An old planet (eg. several billion years older than our own planet) with only a small amount of water coverage and very little tectonic activity - a planet where all the mountains have been ground down by erosive forces - could easily create a world with lots of dust and high wind speeds that could give you a duststorm filled world. However, such a planet might not have a rotating liquid iron core that generates a magnetic field that our planet enjoys. Without a magnetic field the planet would be bombarded by radiation from space.
But to be honest whether an old world like this would have a spinning core or not (and what you could do about it if it didn't) is worthy of a separate question altogether.
Roadside NO2 diffusion tube monitoring with height in London and Cardiff. Each green point indicates the average monitored NO2 concentration for each floor and the red error bars indicate the minimum (left) and maximum (right) NO2 concentrations measured at each floor.
