I don't understand why nobody is recommending the products that are designed for this. Commercial drain cleaners (also called "drain openers") are either based on sodium hydroxide or sulfuric acid. Don't use them together, as they would simply neutralize each other and risk melting your pipes during the process.
You aren't supposed to use sulfuric acid on steel pipes, but it's fine for plastic. The black liquid drain cleaners are usually sulfuric acid. It's stronger stuff, and makes worse fumes than the sodium hydroxide. It's cheaper as well. There might be corrosion inhibitors added, to spare your pipes. And don't use the hydroxide when cleaning up fats/oils, as some will be turned into a soap which is even more solid than the original form. Both classes or drain cleaner can dissolve organic matter like hair, paper, and flesh.
You pour a product into the drain. If the product is powder/flakes, add the minimum amount of hot water you can add to make it dissolve. Wait the amount of time the packaging says to wait, then flush the pipes out with water. Hot water, if that's what the product says to use.
It won't hurt you much to get these products on your hands if you quickly wash it off, but eye protection is mandatory, as you don't want to risk your vision.
If you find the right product, it will cost $2 or less, but I couldn't tell you where to look. A supermarket or hardware store will have a branded drain cleaning product, unbranded might be what you need to look for if you are short on funds.