You'll need to print out some high contrast, black and white negatives, or you can place objects on the fabric for a solar print. Doubling the transparencies increases the contrast. When I used a single transparency, I got a two-tone print, which is nice if you want texture, but not clarity.
I first create a temporary shelter inside so I can work in shadow and not expose the light-sensitive dye too soon. It's fine to do this indoors in the daytime. Paper plate, old brush, cotton cloth, gloves, vinyl tablecloth. This is the brown dye.
Pour out about a Tablespoon or two of dye for a 9" x 12" piece of cloth. You can always add more, but don't pour out more than you can use in the next few minutes. Start painting it onto the fabric. You can tape a border to mask it if you want a clean edge.
Cover the entire cloth. It's still pretty light. (Yours won't be so blurry.)
Place your doubled transparency on top of the wet cloth (I did it toner up/smooth side down so I could clean the back later). I printed out both negatives and positives onto transparencies. To get a crisp print, print out two of the same, align them, and tape them together to make a rich black to block out the sunlight. This is one case where a double negative isn't not acceptable. ;
Pin the transparencies and cloth to a piece of cardboard.
Place in direct sunlight.
This is after 15 minutes in upper 70s F weather. Do these on a bright, sunny day if you can. I like to make sure the cloth dries before removing it from the board. In August, between 11:30am and 4pm left me enough light. Exposure time is 10 to 20 minutes. I like to let it go 20 minutes unless it is a very hot day.
Peel off the negatives. What you see is pretty close to what you'll get.
The dye leaves some residue on the transparencies, which can be cleaned off right away by spraying it with water and wiping with a paper towel. If you've left it there awhile, try soap and water. If you want to reuse the transparencies multiple times, make sure you remove as much residue as possible because it will eventually interfere and degrade the printed image (unless this is the effect you want).
Prepare a little bath. I found a large food storage container was perfect for the basin. Fill it with hot water. For this amount, you need about 1/2 teaspoon of the SolarFast Wash or Synthropol (these are heavy duty detergents to lift off excess dye). For good measure I added about 1/2 teaspoon of soda ash (a fix agent).
Place the cloth in the hot water and agitate it for ten minutes by moving it around, squeezing gently, etc. I found that even with two different colors, they did not bleed onto each other.
Rinse the cloth thoroughly, wring it out, and lay it flat to dry. Easy to iron later, but does not require ironing to set.
Regarding the colors described by the manufacturer: they don't really match their descriptions, as other reviewers have noted, so here's the rundown of colors I've tried and how they revealed themselves to me. Brown, blue, and purple are the only colors I found to be true to their names.
Brown: chocolate or dark wood color with slight oatmeal tint to the white parts.
Sepia (not brown): tawny lion tan with yellow tint to the white parts
Avocado (not dark olive!): bight greenish yellow with yellow tint to white parts
Black (not black): midnight blue with grayish tint to white parts
Teal: green blue with greenish tint to white parts
Red (not fire engine red): pink with light pinkish tint to the white parts
Burnt Orange (not pumpkin orange): dark salmon with light salmon tint to the white parts
Blue: pure dark blue, close to what you would expect from a cyanotype, slight lighter blue tint to the white parts
Purple: deep grape purple, slight lighter blue to tint to the white parts
I found that mixing the red and the supposed black make a more convincing black. So mixing may be the way to get the colors you want. All supplies are available through the wonderful Dharma Trading Company in San Rafael, CA.
Addendum 3.16.18: In between rain storms I have been trying to print. It's useful to look up the UV index in your particular town, day, and a particular time bracket. I found that printing at UV 4 or so meant I needed to let the negatives sit for twenty minutes or more. I found WillyWeather.com useful when I typed in my city.