I don't know where to get it for the cheapest, but I definitely wouldn't be buying it from Mood. Mood may be discount fabrics, but they are discount designer fabrics which are going to be expensive even with the discount. Most of their fake fur is in the $50 a yard range. The only reason I'd buy anything from Mood would be if I was looking for something very professional-quality that I really couldn't find anywhere else.
I can help you with the dye part tho!
Most faux furs are 100% acrylic fabric, or have a large percentage of acrylic and then other manmade fibers in it. These kinds of fibers are hard to dye. You can't just go get a box of RIT and toss it in there. It might turn colors, but it's really only the dye sitting on top of the fur, and it will rub off and make a mess. Plus the color will be pretty dull.
You need to get special "disperse dyes" to dye these kinds of fibers. The most widely available is "iDye Ploly" (specifically the Poly one, don't get the one that just says "iDye" ). iDye Poly is actually meant for polyester and nylon, not really acrylic, but it will dye acrylic a medium or pale shade, but not anything vibrant.
Here's some more information on dying acrylic.