Not easy to extract off a light background and then put on a solid black one, especially with a black subject!
Easiest would be retake the picture
What I'd try is Select > Color Range. Use Selected Local Clusters and Ctrl click on parts of the background using fuzziness and range both around 50%. Try to get most of the background fully selected (it will go white in the thumbnail) without losing too much fur.
Then say OK, and invert your selection.
Then on the mask use Refine Edge. From the drop down list, select the one which shows on a black background so you can see roughly what it will look like. If you have CS5/6 use the edge detection smart radius thing with a radius of 10-20px. Play with feathering and smoothing.
You can also experiment with the "decontaminate colors".
You're likely to always end up with some of that light brown though. The only trick I know to clear that up is to select a color from the fur, probably near black, and use and Inner Glow layer effect, adjust the size until it goes away.
The Layer menu also has some Matting options that sometimes help: "defringe", "remove white matte" and "decontaminate color".
I'd suggest you put it on a darkish brown background, not black, so that the leftover background