What is the best way to attach backing on a quilt with irregular pattern?
This is a perfect example for 'depending on your taste'.
I agree with you that vertical and horizontal lines don't do justice to the pattern, but then again, others may disagree.
That said, if I were to sew backing, I'd follow (some of) the contours of the cat.

If the backing is black, would it be better to use black thread to go over along the edges of all the pieces when connecting the backing? Or, should i use the same color thread as on one of the neighboring pieces? 
If you're sewing with a sewing machine, you can use two different colours of yarn, one for the front, one for the back. When sewing by hand... I'd choose a colour matching the front side for that's the side you see most. And make sure the stitches at the back side are as small as possible so the cat shape would be more apparent than the colour of the thread.

Are there any other (better) ways to do this? Is it possible to not attach the backing at all, other than around the perimeter or would that not work so well?
Depends on the size of the quilt. But since this is a free form quilt pattern, why not attach the backing free form as well? Either by random 'squiggles' or a more regular pattern. Do an internet search for 'quilt free form backing pattern' to get some inspiration. You could also just fill the black background with a free form pattern and follow the outlines of the pieces that form the cat itself. Or use a different (free form) pattern for the cat. The possibilities are endless