I love photographing cats:
Mine:
Tonks
Shelter cats (I volunteer as a photographer for one):
Adoptable Cats at Ollie's Place - frogoutofwater
Other peoples' pets:
Other Peoples' Pets - frogoutofwater
Random cats on my travels:
Borneo and Singapore Adventure 2014 - frogoutofwater
And big ones:
Zoo animals - frogoutofwater
If your cat wants to come close, it sometimes works to stroke the cat with one hand while snapping a photo with the other (experiment with a macro lens - the first one was taken with a LensBaby):
Tonks Adoptable Cats at Ollie's Place - frogoutofwater
Wait for the golden hour (in cat photography, that's when kitty is sleepy enough to stay still but not so sleepy that its eyes roll back into its head):
Tonks
Get her a box or some other container, set up your shot and wait:
Tonks
Let her play, and wait until she takes a break (I feel like she's saying "was it good for you, too?" in this shot):
Tonks
Give him a toy or a prop:
Adoptable Cats at Ollie's Place - frogoutofwater Tonks
Sing to them (I've heard harmonica works well, too):
Borneo and Singapore Adventure 2014 - frogoutofwater Tonks
As for focusing, try focusing on an ear at the point where there is some contrast with the background. You might have to try several times (because sometimes the camera will focus on the background, not the ear). It can also work to focus on the point in the eye where you get contrast between the eye and the black fur. I try to use at least f/4 (preferable f/5.6) to get enough depth of field to have the near eye and nose in focus on a cat. With a black cat, it also helps to photograph against a mid-toned but contrasty background (as opposed to something very dark or very light) - blue, green, mid-purple or red for example can work well.
I have used flash with cats but I dial it down a bit and put a Gary Fong light diffuser on it.