I've seen when Kakashi was training Naruto with his elemental affinity that Naruto cannot see what his clone is doing until it is undone. How does he control the clone? Does it move and think like a separate entity? Can it think to try to kill the original?
Does it move and think like a separate entity?
Yes, the clones are completely independent, apart from the fact that Naruto can undo any clone he wishes to.
This can be seen when they collaborate and the fact that they need to communicate in order to do so. Note that when Naruto creates them, they know what he knows so he can think of a plan, create the clones, and everybody knows their role in the plan without a need to discuss it—they are still independent.
Can it think to try to kill the original?
Possibly. But seeing as they have a short life and have just been cloned, Naruto would have had to be thinking about that kind of thing himself already in order for a clone to really do that. And of course, since Naruto can undo a clone at will, it would mean that as soon as he senses a threat from a clone he can undo it.