Well, I don't know of a pattern, but there are a couple ways you could go about it. You could knit a tube or you could knit a flat piece and then seam it together.
To get it the perfect size it would be best to do it like you'd do a garment for yourself. That is knit a swatch, count the stitches per inch, measure the dogs leg and multiply to see how many to cast on. However.. it IS a dog and a small one at that cast on some stitches knit for a little bit and wrap it around her leg and see if it fits. If it's too big try again, etc. It's good practice anyway.
If you want to do a tube you'll need to have double pointed needles in the correct size for the yarn, or two matching circular needles, or one long circular needle for magic loop. Since you're learning though I think knitting flat and seaming is the easiest.