orbiting a camera around its focal point

To rotate an object (camera, actor, light) around a fixed point, in 3D space, you need determine the 3 coordinates for the moving object. The comment from grg99 assumes the camera rotating around focal point by maintaining a fixed z value, that is, the camera will rotate around a line parallel to the z axis, passing by that point. The z coordinate can be any fixed value as you want, which can be smaller, equal or greater than the z of the focal point, depending on where do you want to put the orbit.
If you want to rotate the camera around a line parallel to the y axis, then just sustitute y by z in the formula and use a fixed y value.
You may want to rotate the camera around the vertical line (parallel to y axis) and, at same time, make the camera to move up and down, so you can vary also the y value, instead of use a fixed value.
Finally, if you want an "inclined" orbit, not parallel to any plane (say, XY, XZ and YZ), then you should use spherical coordinates to determine the 64 camera positions, if you want a circular orbit.
For a camera C with an eliptical orbit, inclined 45% related to to XZ plane (commonly accepted as the "horizontal" plane),
around point P, at an average distance of R between C and P, you can do the following algorithm:

step=360/64
for position = 0 to 63
{
   angle = position * step
   C.x = P.x + cos (angle) // essentially the same stated by grg99
   C.z = P.z + sin (angle)
   C.y = C.y + sin (angle)
   look at point P
   render the frame
}

Jose