
Signing up with some canine model agencies is a good start. You will ned lots of good photos and to be able to write a kind of "CV" for your dog.
I have not done film work with mine but have known people who do work with other animals, and as regard training the dogs must be well trained (of course) but teaching things like a good sendaway, and targetting, are absolutely essential, as those are the basics for getting a dog from A-B on set.
They should also be very good around other dogs (which ruled mine out when applying), and also totally bombproof in unfamiliar surroundings... they MUST take everything in their stride. That's a big consideration as many dogs get anxious about the fact that on set, there are swinging lighting and mic booms above their heads, not to mention huge cameras moving about with people sitting behind them, and sometimes machines that blow wind, make rain or snow.
They also have to be very social dogs with a genuine ability to get on with people, as they would be working with actors, strangers, all kinds of background personnel, plus also of course some people wearing very weird and scary costumes. They also need a BIG off-switch as most of the time working on set is waiting around - you could wait there for 8 hours and not be called. A restless dog in the background whining and wanting attention is a liability.