Napoleon had been very sly from the outset. It is clear from the differences in opinion between him and Snowball that he saw him as a threat. Since he wanted sole power, he secretly trained Bluebell and Jessie's nine puppies in a loft after having removed them from their mothers once they were weaned. No one realized what his plan was with them at the time. It soon became apparent that Napoleon had been training and indoctrinating them to become his enforcers and security guards. They were taught to respond to signals given by him and were, obviously, loyal only to him. He was their master and had exclusive authority over them. Napoleon was biding his time to put his aggressive minions to good use.
The opportunity arose at a crucial time. Snowball had developed plans for the building of a windmill. He was passionate about the project for he believed that the farm would benefit from one. Napoleon, however, had rejected the idea from the outset and showed his disdain by urinating on Snowball's plans. The animals were to vote on whether to build the windmill or not and were gathered to hear both Snowball's and Napoleon's speeches on the topic. Snowball spoke first and gave reasons for building the windmill. Napoleon then mentioned that the whole idea was nonsense and suggested that no one vote for it.
When Napoleon said this, Snowball jumped up and made an eloquent and passionate speech about the benefits of having a windmill. So convincing was he that it was clear that the animals, who had been undecided up till then, would vote in his favor. At the end of his speech, Napoleon got up and gave a high-pitched whimper which no one had ever heard before. Once he had made this sound, nine huge dogs came bounding up. They immediately headed for Snowball who, in fear for his life, ran as fast as he could to escape them. He ran across the pasture towards the road. There was an occasion when he slipped but he regained his composure and ran as fast as his trotters could carry him. Just as the dogs seemed to be gaining on him, he slipped through a hole in the hedge where the dogs could not follow and escaped. Snowball was never seen again.
It is clear that Napoleon had plotted to get rid of anyone who opposed him once he started training the dogs. Once he had the dogs completely in his control, he could use them to terrorize and kill whomever he pleased. This is exactly what he did and his actions with regard to Snowball signify a dramatic turning-point in the novel. This was the moment in which Napoleon displayed his might and would mark the beginning of his tyranny.
Napoleon does not really do all that much to chase Snowball off the farm. Instead, he relies on his dogs to do it for him.
At one point, Snowball is speaking at a meeting. Napoleon decides it is time to get rid of him and he emits a high pitched sound. That, apparently, is the signal for the nine dogs to chase Snowball. Snowball runs for his life and leaves the farm.
So all Napoleon really does right then is to emit the high pitched sound. Other than that, it's just that he had trained the dogs.
From the very beginning there has always been enmity between Napoleon and Snowball. This enmity reaches its climax in Ch.5 when the animals hold a meeting to decide on Snowball's proposal to build a windmill.
Napoleon has always been jealous of Snowball and he is opposed to the building of the windmill. In fact, while all the other animals were appreciative of Snowball's efforts in building the windmill, Napoleon urinated over Snowball's draft plan of the windmill.
In Ch.5, all the animals gather in the big barn to vote on whether or not the windmill will be built. A heated debate arises which finally ends with Napoleon emitting a queer sound. At this signal nine huge dogs rush in and chase away Snowball leading to his permanent exit from the farm:
How and why do snowball leave the farm?