To cycle a tank you need to grow denitrifying bacteria to consume ammonia and nitrite that your tank produces. The bacteria needs an ammonia source to grow colonies sufficient in size to consume all the ammonia and resultant nitrite and turn it into nitrate which typically you remove through your regular water changes.
A fishless cycle uses an ammonia source to replicate the fish waste that a tank of fish would produce. This ammonia source can be pure ammonia, an aquarium specific ammonium chloride product like Dr Tims Ammonium Chloride, a cocktail shrimp or fish food.
Ill assume we are using an ammonium chloride product.
Set up your tank. Make sure everything is running smoothly. Make sure you have used a water conditioner product with any tap water you have put in your tank. If you have an adjustable heater raise the temperature to 28c/82.5f.
You should have a test kit. Preferably a liquid test kit. It should test for ammonia, nitrite, and nitrate.
Dose the ammonia chloride to approx 4ppm and start testing daily for ammonia. Once your ammonia drops below 1ppm redose it back to 2ppm. This may take a couple of weeks.
Start to test daily for ammonia and nitrite. Whenever your ammonia drops below 1ppm redose it back to 2ppm. Nitrite causes false positive nitrate readings, so no point in testing for nitrate until your cycle is complete. False positive results will just confuse matters.
You should start to see nitrite and in your daily tests. Over time your nitrite should start to rise and the amount of ammonia should start to drop further. Your ammonia may start to not be detectable in your daily tests. Keep redosing ammonia daily if you see it below 1ppm. Your nitrite may rise off the testing chart. I prefer to keep nitrite within measurable levels so it shouldn’t hurt to do a water change to keep readings on the chart. Remember to add water conditioner whenever you put tap water in the tank.
Over time your nitrite should level off and begin to fall in a similar manner to what your ammonia tests did. When you are able to dose ammonia to 2ppm and 24 hours later see 0 ammonia and nitrite you are cycled. At this point you have enough denitrifying bacteria to consume all the ammonia and nitrite of a moderately stocked tank. You may want to continue dosing ammonia for a few days to make sure it continues to consume all the ammonia and nitrite and be sure your cycle has properly established before proceeding.
Your nitrate will likely be very high, so now test for nitrate. Do a big water change to get this down. Preferably below 10ppm. Adjust your temperature to the needs of your fish. Get your fish, acclimate and add to your tank. I would advise stocking lightly to start with and slowly adding fish until fully stocked.
A fishless cycle typically takes 6 to 8 weeks.
A good way to speed up this process would be to put a small amount of filter media from an established filter into your filter, or get a sponge from an established filter and squeeze it into your tank water. Perhaps you have a friend who keeps fish who could let you have some? This will seed your filter with the bacteria you are trying to grow and speed up the process.
Another option is bottled bacteria like Dr Tims One + Only or Tetra Safestart. These products wont instantly cycle a tank as they claim but in a similar manner to adding established filter media they can seed your filter with the bacteria you are trying to grow to establish your cycle. These products are hit and miss as to whether they work at all, but are an option if established filter media isnt obtainable and may speed up the process from several months to several weeks.
Stores have to sell for more than online because it costs more to have that brick and mortar store than it does to have a website. What you are paying for in the store is your ability to get things NOW, advice from the shop and the ability to see you are getting healthy specimens or undamaged stock. You don't think about those things until your local shop closes and there's nothing that takes it's place. The first time your fish are sick and you are not going to get your item from online for 24-48 hours, for some diseases, that's already to late to save them. 
