Where did the ninjas come from?

The ninjas are not an organized group like the samurai. They do not adhere to a strict tradition. They are mercenaries and assassins working at the whim of the highest bidder. Unlike the samurai who will be lost to western ways, the ninja can work with western ways to keep getting paid.

They were not sent by the Emperor, who despite being young and a puppet, still does not want the samurai killed. They were sent by the Japanese business man who is the real antagonist of the film, Omura. He is trying to push the westernization and modernization of Japan by any means possible and see the samurai refusal as a barrier. He attempts to send the untrained army that Tom Cruise is supposed to train to attack the Samurai when the samurai attacked his train line. He also controls the emperor.

The ninjas, in the movie The Last Samurai, were bounty hunters. The costume design was obviously taken from the design from the modern vision of the Edo ninja which would be why it might be confusing.

Pure black clothes with leathers was never used in Japan like what is shown in the movie. The kind of clothes that were used for night assassination attempts were usually dark blue or gray clothes darkened with soot. Black clothes were actually really pricey because they required quite a lot of black ink to produce, hence why nobody would use it for something like assassination. (You could easily find out who did it by looking around for someone or a group who purchase a lot of black ink.) This is why you may find "some" drawing of people wearing jet black clothes, but they are usually people with high prestige. (The history of color in fashion is quite interesting. As an example, in Europe for centuries, England took the red because it was the 2nd most pricey color to produce, after a certain blue color which could only be produced with a certain rare rock from the Saudi Arabia. France controlled the flow of that rare rock in Northern Europe and applied such an heavy tax on it when it crossed toward England that England's king decided to ban the blue to cut the exchange of finance toward France. In Japan, the black was an equivalent to this effect as it required a certain dense ink in great quantity to get the jet black finish. This ink was produced with a mix of Obsidian dust and octopus ink unlike the writing ink which was produced with a bit of oil in it.)

In matters of facts, we shouldn't really call them ninja. As ironic as it might sound, a "ninja" back then couldn't be anything less than someone of a relatively good status, hence never "nobodies" like shown in the movie. All the materials to learn to become a "ninja" (politics, biology, anatomy, combat, reading, mathematics etc.) was way out of any peasants hands. Usually, those who could become "ninja" were the youngest children in a house of high status. While the oldest boy was trained into combat and politics, the younger were trained into assisting the oldest in as many ways as possible.