Probably Something Subterranean
There's a lot to consider, but right off the bat living underground could really help. Your survivors need someplace to hide and depending on the climate (which can be completely controlled by the aliens) protection from the elements. Living underground could allow the survivors to use hydroelectricity or geothermally generated electricity without getting caught which allows them to grow crops using artificial light (once again underground where the aliens can't see it). If this sounds too high tech their underground dwelling can be a cave system with underground rivers and or geothermal activity. It's only a matter of converting already common generators and either some super sketchy makeshift incandescent bulbs or finding a bunch of LED bulbs (it's still crazy hard, but at least somewhat believable). They can also use all of those dead trees to grow mushrooms. If the ocean is nearby and most of the Earth is not covered in smoke, fishing is also an option.
The Smoke Opens up All kinds of Interesting Options
The "smoke" was created by aliens with technology well beyond ours. 197
This means the "smoke " can have all kinds of properties to suit your needs. You would expect black smoke to absorb more thermal energy from the sun and raise temperatures, the aliens are probably expert geoengineers and can use additional layers of aerosols above the "smoke" to balance out the additional absorption or even make it cooler. The "smoke" can also have handwaving properties like perhaps it makes things look dark, but the sunlight still reaches the Earth's surface so growing crops and the water cycle still work as usual. Does the "smoke" negatively affect breathing like real smoke? Maybe it does the opposite and people feel more alert. Overall, once you fill the air with some mysterious substance created by aliens, you have unlimited possibilities which can shape your encampment in all kinds of ways. You can figure out the plot, motivations, and character development then tweak the properties of the "smoke" to accommodate them. In practice, the behavior of the environment doesn't need to be justified because it's literally aliens. The main thing is to establish clear rules early on so readers can follow along. 279