Hirohito is probably the most extreme example of this deificaiton, and the belief that the Emperor was a living god had a great effect on Japan's actions in 1937-1945 - But were the previous post-Restoration Emperors, Meiji and Taisho, deified to a similar extent?
To extend this, was there a pre-Restoration precedent for the idea of a divine Emperor?
Lastly, how widely would it have been believed that the Emperor was divine in the upper echelons of the Japanese Administration? Would people such as Tojo and Yamamoto, for example, have really thought the Emperor to be a god?
The emperor has been considered divine from the beginning. Namely, the line of emperors has been considered to be descended from the grandson of the sun goddess Amaterasu. I'm not sure how one can say it really had a great impact on Japan's actions in WWII: however: Japan's foreign policy was almost entirely directed by natural resources, poor government control, and nationalism (or the so-called "kokutai").
Technically, the emperor has always been the absolute leader of Japan. However, various noble families from time to time have exerted strong political control over the emperor and by extension the Japanese imperial government. After the emperor established the title sei-i-taishogun, or "Supreme Barbarian Conquering General," various influential figures received the title and used it to create a series of shogunates that essentially ruled while treating the emperor as merely a figurehead.
It is also important for one to understand that the Japanese kami, or "god," is not quite the same implication as that of a Western "God." Rather than being some sort of all-powerful being, a kami is a deity that represents some sort of aspect of life. For instance, there is often considered to be a kami in many basic common things, like things like kami that haunt those who do not honor the dead, to a kami of traffic safety. There is also a kami at many of Japan's shrines, where offerings to the shrine are in spirit offerings to the kami of the shrine. In the case of the controversial Yasukuni shrine, the kami of that shrine is supposedly composed of the spirits of the Japanese war dead.
To this end, the reason why the emperor was viewed as a "living god" was not that he was some sort of all-powerful ruler, but more in line with the Chinese concept of the Mandate of Heaven and the European one of the Divine Right of Kings-namely, the emperor is the legitimate ruler because he is divine, not that he would smite the enemies of the Japanese emperor with supernatural force.
While I do not have direct quotes from either Tojo or Yamamoto as to their stance on the Emperor's divinity, both of them routinely went around the Emperor's orders if they found it convenient, and often times they excluded the Emperor from pertinent tactical information in Imperial Conferences, for instance re: Guadalcanal.