I'll have to say photographing cats is a lot easier when you have a good
deal of experience and photographic equipment. Most often, I take
pictures of mine by natural light, and get quite close. Natural
lighting, of course, avoids eye reflections.

The minimum thing you can do to avoid reflection and still use flash is
to use some sort of extender that lifts the flash above the central axis
of the lens. It may also work, if the flash head is movable, to bounce
it off a light colored ceiling, or even a piece of cardboard.

In my preferred setup, I use one or more free-standing studio flash
units.  There is a wireless remote trigger on the camera, with a
receiver on one of the flash units. The other flash units trigger when
they sense the first flash going off -- the delay is thousandths to
ten-thousandths of a second, so even cat movement isn't a problem.

The setups I am going to describe can also be done with much cheaper
floodlights, which also give you the advantage of letting you see the
lighting effect. Professional flash units have low-power modeling lights
that also serve as a guide.  Many color negative, but not slide, films
can tolerate the yellowish tinge of ordinary light bulbs rather than the
pure bluish-white of sunlight.

A minimal setup for human or feline portraits uses a single light source
high above the camers (at least 3 feet/1 meter), preferably diffuse. I
usually diffuse by reversing the flash and aiming it at the inside of a
metallized or white "umbrella" reflector.  You can get a bit more
shaping by moving the light perhaps 30 degrees to one side or another of
the camera axis.

If you have the luxury of additional flash units, and a cat that will
stay fairly still, putting a small flash directly behind the cat, so
that the cat's body blocks the direct flash into the lens, will put
something of a halo around the body edge and make fur stand out.  
Alternatively, you can put a second light source at right angles to the
cat. Try to have this about half the intensity of the main light, or 2-3
times farther away.

Other lighting can help, but you are really getting into extensive
equipment, and someone that invests that much probably knows how to use
it without my advice!