Vertical farming refers to growing crops in vertically stacked layers (thus being able to grow more plants on a same space). It can use simply containers with soil, hydroponics, aquaponics, aeroponics, ...
Containers with soil
Containers with soil allow plants to get their nutrients from the soil itself, thus not requiring (expensive) nutrient fluid. The soil can be replaced every year (or sooner) with other soil with more nutrients (for example obtained from a compost pile). Easiest to do this is with gutters (perforated pipes are more difficult for this).
Some often used (low cost) techniques include using pallets and cloth (to make pockets on the pallet to plant in), or (plastic or aluminum) gutters or perforated plastic pipes. Gutters can be placed on an A-frame (which can be made from timber, or wooden or aluminum gridbeam or thick bamboo). See some example here:
A simple A-frame can hold up to 12 gutters or perforated pipes:
Whatever the material is used for the gutters or pipes, all pipe/gutters can be placed on a slight elevation to allow excess water to flow out. All of them can either be fitted with a drain pipe to drop the water straight into a container and the container can be fitted with a simple aquarium pump to pump excess water from that container back into pipes/gutters, thus recycling the water (and some nutrients) and reducing the water requirements. The pump can use a time switch and the duration of continuous watering can be set per day (thus making sure the soil isn't overwatered, which would cause plant roots to rot).
If using plastic pipes or gutters (which may be the easiest to acquire and relatively inexpensive), you could use a geotextile cloth in the pipe to further reduce the chance of pollutants from the plastic to leach through into the plant itself (via the soil or water).