What is the best way to plant St. Augustine grass?
This is difficult to address because it sounds as if you're giving the grass everything it needs. It's possible that there's an unusual problem. If so, you'll need the help of a local expert, such as a county extension agent. The amount of fertilizer and water you give your new grass plants could be a factor. Although the grass needs little fertilizer to get off to a good start, too much may cause the tender new roots to burn. Be certain to use no more than the recommended amount.
Watering of new plantings can be tricky. Until new roots become established, watering will be needed more frequently than for established grass. On the other hand, excess water could cause the roots to rot. To complicate matters further, weather conditions affect the frequency of watering and amount of water needed. During dry, sunny, or windy conditions, more water will be needed than during overcast, humid, or calm weather. If your site is sunny (no more than a few hours of shade each day), try a different grass, such as zoysiagrass or Bermudagrass. If those die as well, have a sample of your soil tested to see if there's some sort of contamination. For example, some topsoil is taken from agricultural land and may contain residues of weed killers. The only thing you didn't mention was sunlight. Is your 'small backyard' extremely shady? If so, maybe grass isn't the best choice. Check your local nursery for ideas on shade-loving perennials, ground covers, and shrubs.