Allow the newly cleaned tent to dry completely, and voila! A clean tipi. Or cleaner, anyway. Don't expect miracles, particularly with some of the finer varieties of dirt or with oil-based stains. Dust that is fine is very difficult to remove with an ordinary soap-and-water treatment, and it may be helpful to first vacuum --not brush--as much dust as you can off the fabric before commencing with the washing operation.
To wash your tipi, you need to have a warm, rainless period of sufficient duration to allow the tent to dry completely.
Now for tools.
My general rule for soap is not to use anything that I wouldn't mind having my hands in for a while. Mild detergents intended for hand-washing dishes are ideal. Anything stronger might damage the various treatments that the cloth was given at the mill.
The best brush is one with soft bristles and a long handle. They are intended for washing cars without damaging the paint; you find these in the automotive section of the larger discount stores.
For what it's worth, my current brush is called the Hoppy Brush and was bought at my local Kmart for about twenty dollars. It has a telescoping handle and can be connected to a garden hose, although that makes the brush very ungainly to use, and the small size of the built-in nozzle constricts the water flow to a puny trickle.