The entirety of this answer is from CottonAcres.com, a website with loads of information, that crucially sells cotton seeds.

Growing cotton in your garden
(The website also has instructions for growing cotton indoors.)

Cotton is a perennial plant but mostly grown as an annual as it cannot withstand frost. Cotton needs a lot of sunshine, warm conditions and 4-5 months of frost-free temperatures to mature and produce cotton.

1. Work the soil to 3-4 inches deep to eliminate weeds.

2. Incorporate compost into the soil to provide nutrients for the cotton plant.

3. After all danger of frost has passed and soil temperatures have warmed up to 60 degrees, its time to plant cotton seeds.

4. Plant cotton seeds in moist soil, in groups of three seeds, 1 inch deep and 6 inches apart.

5. Cotton seeds germinate in 7 to 10 days if soil temperatures are above 60 degrees Fahrenheit (16 degrees Celsius).

6. Cotton does not like a lot of water. So water once in about 7 days. Water more frequently in summer months if leaves show signs of drying.

7. Feed the cotton plants with a granular or liquid plant fertilizer like a Rose Food available on the market. Follow the fertilizer dosage instructions on the label.

8. Large yellow flowers appear in 45 days from sowing. Flowers wither off and form bolls.

9. Stop watering 16-18 weeks from sowing when bolls have been formed, so that the plant begins to dry and shed their leaves, and the bolls will split open to form a fluffy ball

Note: Plant basil, onions, anise and coriander to repel flies and worms. Invite birds to your garden using bird feeders to keep cotton pest free.

Source: http://www.cottonacres.com/grow-cotton.html