I don't think there are any optical elements in the 'cheap toy' type of
kaleidoscope; just a sheet of shiny metal or plastic bent to form an angle,
and a shallow tray of 'interesting shapes and colours' with a translucent
backing.  You see a pattern formed by the directly-visible bits of stuff
and their reflections (and possibly reflections-of-reflections).  An
auto-focus camera  may struggle to work out where to focus; the distance
from the 'eye-hole' to the end of the instrument, would be very slightly
more than the actual distance that needs to be focused on.  That may be
too short for some cameras to manage - the 'macro' setting may work
better, if there is one.  

If you are prepared to sacrifice the kaleidoscope, it may be easier to get
photographs of the patterns by arranging the tray of bits and the angled
reflector in front of the camera, with an opaque covering to enclose them
and the camera so that light only gets to the camera via the tray of bits
and the reflector.  You could then arrange distances to suit the lens.
Getting the alignment of the parts, including the camera, 'just so', could
be tricky.

A web search for

   kaleidoscope photography

should yield some interesting links  ;))

Kaleidoscopes themselves can easily become something of an obsession; be
warned!