This is an old thread, but I hope it can save somebody a lot of time.
While fur can be simulated with hair particles, it is possible to get fur textures a couple different ways. One could be simply opening Gimp or Photoshop and cropping a picture of a dog so that only a fur pattern remains. Another is to render a plane with a particle system that closely resembles fur.
After this step, you will have a picture of fur, but it probably won’t look good when it’s wrapped onto a 3D model. However, you can set the size and direction of the fur on the dog’s material texture by projecting it onto your 3D model with stencils. Stencils can be used to apply a texture to the dog’s material texture with the texture paint mode, or they can be applied to the actual 3D model if its poly count is very high.
I chose to create a fur texture generator for my own dog model. I didn’t end up using it because I preferred the particles, but I’ll share it here so other people don’t have to waste time setting up the same system. The generated textures are here because they’re too big to post on this page, but the source .blend file was small enough that it could fit on this page.
I hope this helps!
FurBrushGenerator.blend (897.1 KB)
Sources:
https://docs.blender.org/manual/en/latest/sculpt_paint/brush/texture.html