When it comes to making dreamcatchers, tradition would dictate that you use willow hoops, animal sinew or leather strips and perhaps a sacred feather from a bird of prey. Since most craft stores don't carry such things, you might consider a modern and purely decorative twist on the dreamcatcher like this lace doily version.
Don't mind going non-traditional with your dreamcatcher? No problem. Grab a large brass ring (for sale at nearly all craft stores specifically for dreamcatcher crafts). Select a neutral-colored yarn, a couple of colors of cotton thread or embroidery floss, a plastic yarn needle and a lace doily.
Wrap a neutral-colored yarn in a tight uniform layer all the way around the large brass ring. When you get to the end of the ring, tie a knot with the yarn and cut off the excess.
Cut a piece of thread or floss to a length that is two times the circumference of the ring. Tie a knot at the top of the brass ring with the string and thread the other end through the yarn needle. Send the needle from front to back around the ring to wrap the string around it, and then pull it back towards the front, crossing the string over itself. Continue threading the string around the ring in this manner, spacing the loops about an inch apart. Tie a knot after the last loop and trim excess string. Cut another piece of thread/floss in a new color and tie it to one of the loops you created with the first string color. Thread the other end through the yarn needle and continue looping it around the first set of loops. Continue weaving until you reach your desired web size. Keep in mind the size of the doily when creating the web -- you'll need your doily to fit in the center of the web. Tie a knot when done and cut off excess string.
See this dreamcatcher tutorial for more detailed instructions on weaving a dreamcatcher web.
Cut a piece of thread or floss to a length that is two times the circumference of the ring. Thread it through the yarn needle and use this to "stitch" the edges of the lace doily into your web. You can adjust the tension of the doily as you work, or after you've secured the doily into your web, making it as slack or taut as you'd like. When the doily is completely woven into the web, knot your thread and trim any excess.
Tie yarn tails, feathers, beads or other decorative items to the base of your lace doily dreamcatcher. Generally, the tail length is twice as long as the diameter of the ring, but make it any length you'd like. Bear in mind that using a doily is in no way traditional, but it does result in a lovely dreamcatcher.