I'm not too familiar with the shogunate-imperial dynamic during and directly after the Onin War [1467-1477], but I can give you a good idea of how the shogunate consolidated and maintained power from roughly 1600 and beyond. Additionally, this post will not necessarily be just about how the shogunate staved off the emperor, but rather how the shogunate generally managed to maintain power over a number of centuries.

In 1555, Oda Nobunaga began his conquest of Japan, often resorting to brutal and ruthless means in order to do so. With the emperor's power weakened because of the destruction of the Onin War, Nobunaga had few military threats to confront and was essentially free to unleash his armies at will. By the time he was assassinated a little less than three decades later, Nobunaga had conquered over 1/3 of Japan, while simultaneously building a strong and secure alliance of powerful daimyo. To ensure that his rule would be stable, Nobunaga enacted reform of domestic political institutions which would later allow his successors (the Tokugawas) to establish and maintain peace.

Nobunaga is particularly important because he set the stage for Toyotomi Hideyoshi and the Tokugawas to seize absolute power of Japan. Toyotomi is famous for his military conquest of the rest of Japan, but the Tokugawas are famous for putting into place the social and political systems that made it possible for shogun rule to triumph over the imperial court until the Meiji restoration. It is important to note here before continuing that rulers in Japan were often only as powerful as their followers. In controlling the daimyo, one could control the entire of Japan. This realization is important as we continue into the pre-Meiji era.

Tokugawa Ieyasu, the first Tokugawa shogun, realized the need to control the daimyo; Ieyasu required all daimyo to swear oaths of loyalty and he forbade them (under penalty of death) from constructing alliances with each other. Ieyasu's successors continued this trend, seizing control of unfaithful daimyo's land and redistributing said land to the allies of the Tokugawa. Political opponents were stripped of all but their title and banished to the far reaches of Japan to rule over small hamlets.

However, the most useful tool that the Tokugawas used to disenfranchise the imperial court was the system of alternate attendance, which was enabled by the changes that Nobunaga and Toyotomi had already put into place. The wife and children of all daimyo were required by law to reside in Edo (modern day Tokyo), while the daimyo rotated between Edo and their province every other year. The consequences of this practice are fairly obvious. First, the shogunate basically held the families of the daimyo hostage -- incentive to do as the shogun said and not misstep. Second, the annual daimyo processions from the provinces to Edo cost an enormous sum. Some daimyo even spent even more than necessary on their processions in order to look good to the peasants, at the expense of their bank account. Third, alternate attendance directly weakened political support for daimyo from their respective provinces. This didn't necessarily apply to the first generation that went through alternate attendance, but the children of daimyo would have grown up in Edo, not stepping foot in their home province until they became daimyo themselves. Peasants essentially became ruled by someone that all but was an alien -- this drastically decreased the chances of a daimyo organizing a loyal army at their backs. In short, the daimyo were neutered by the threat of their families being executed, the emptying of their coffers, and alienation from their province.

All this resulted in a concentration of power at Edo, and a significant lack of threats from the countryside. Any peasant or samurai uprisings were swiftly quelled by the powerful Tokugawa armies, the latter of which were backed by the accrued riches of the shogunate. This dynamic continued until the Bakumatsu, when samurai toppled the Tokugawa order in mid-late 1800, but that's another story.

(Sorry to cut it off early -- I should elaborate on how the context changed by the time of the Meiji restoration and why the system eventually fell, but it's quite late where I live and I need to sleep.)

Sources: History major, A Modern History of Japan - Andrew Gordon

There were three eras of government that called Shogunates, and the answer is slightly different for each. Also Imperial authority was seriously weakened even before the rise of the Kamakura Shogunate (the first Shogunate).

Late Heian Period: The Heian period is remembered as the zenith of Imperial Court, but it also saw the decline of actual Imperial authority. This happened for two reasons. In the Capital, the aristocratic Fujiwara clan manipulated the marriage system to marry one of its members to an underage Emperor and have a Fujiwara declared as Regent. The Regent would then act as the head of state. before the Emperor could assume actual political authority, he abdicated, became a monk, and declared another underage boy the Emperor. The Fujiwara would take over the regency for the new Emperor. This isn't exactly what happened every time, and some Retired Emperors remained active in court and could still exert authority, but the general trend was away from direct Imperial rule.

Concurrently, central authority in general was on the wane. Shōen, or fiefdoms, were growing in territory and authority. A shōen is like a fiefdom held by a landlord, but it's a bit more complicated than that since legally they were defined as holdings in which certain people had certain privileges (shiki). Over the course of the Heian period these privileges expanded, and the nobles or temples which held them essentially became mini-heads of state that held essentially absolute authority within the shōen, and often immunity from taxation.

edit: I should note that "nobles" in this comment always refers to people who were ultimately part of the Imperial Court. The people (mostly warriors) later on who were granted authority and land by the Shogunate (or who already had authority and land and had it recognized by the Shogunate) are not nobles (unless they were also granted a title by the Imperial Court).

Concurrently with this (perhaps I should have put three reasons earlier, but it's hard to separate this from the rise of the shōen) was the growing power of the warrior class ("class" used in the informal, descriptive sense, not anything like the legally-defined samurai class that came later). Various border wars in the Kanto region resulted in the prominence warrior clans, clans which were often arose outside the existing structures of nobility and shōen. During the later Heian warrior clans did begin to be granted shōen privileges, but there was still somewhat of a divide between the Heian ancien regime and the warrior clans.

These conflicts turned bloody in the Hogen (1156) and Heiji (1159) conflicts, when the Taira and Minamoto warrior clans became involved in Fujiwara internal politics. The Heiji conflict basically destroyed Fujiwara authority, which was replaced by Taira hegemony. It also resulted in the expulsion of the rival Minamoto clan and the death of its leader. The Minamoto heir, Yoritomo, was spared and exiled to the north.

Rise of the Kamakura Shogunate: The Taira rose to power and tried to become the new Fujiwara, but this didn't work out and after a few decades the Genpei War broke out. This arose for various complicated reasons, but it evolved into a struggle between the Taira and their clients, and those who felt snubbed by the Taira. The anti-Taira faction clustered around Yoritomo.

Yoritomo started the war as a relatively modest landholder in the north, and ended it as the de facto ruler of Japan. This is important because it means that the organization he used to manage his original landholding (the Samurai-dokoro) was expanded to manage everyone who joined his cause, and eventually every major landholder in the country.

Because he was fighting a war, grants of land also became tied up with pledges of military allegiance, so if you were a military valuable ally, you were likely to get a nice grant of land from the Samurai-dokoro. Conversely, if you wanted more land, Yoritomo expected an oath of military service in return.

Although Yoritomo did "accept" various offices from the Retired Emperor Go-Shirakawa once he'd won (including Shogun), the more real source of his political authority was his ability to grant or revoke land through the Samurai-dokoro and associated offices (the Kumon-jo and Monchu-jo), and the related ability to sic angry warriors on you if you stepped out of line (because land ownership was now tied to military allegiance to Yoritomo).

So in this new system, the major criterion for political authority was not "are you a living god," but "can you give me land." The Shogun could, the Emperor couldn't.

(More precisely, it was "can you give me land privileges," it wasn't quite as simple as outright owning land and everything on it, at least not yet)

However, even then Imperial authority hung on: Emperors continued to appoint governors who ruled in tandem with the military authorites granted the same provinces by the Kamakura Shogunate.

The rise of the Ashikaga Shogunate: This system collapsed, and a rebellion of land-hungry Kamakura vassals overthrew the Kamakura Shogunate, and briefly (for 3 years) restored direct Imperial authority in the Kenmu Restoration. However, the Emperor Go-Daigo failed to set up a new regime that pleased everyone and Ashikaga Takauji, a major figure in the rebellion that set up Go-Daigo in the first place, rebelled, set up a rival Emperor and mostly dethroned Go-Daigo. The reasons for the failure of the Kenmu Restoration are complex, but we should note that the rebellion to restore Go-Daigo was led by landholding warriors hoping for more land, Go-Daigo's government favored court nobles (not warriors), and the rebellion to unseat him was led by landholding warriors hoping for more land.

I say "mostly dethroned," because this ushers in the Period of the Northern and Southern Courts, where Go-Daigo's dynasty set up an Imperial court in the South, and Ashikaga's puppet Emperor's dynasty set up another one in the north. The Southern Court had very little real power, though, so I'm going to skip it.

The Shogunate set up by Ashikaga and his successors was less of a solid central authority than the Kamakura. Whereas Yoritomo had been able to purge any landholder who threatened him (and/or take their land), Ashikaga was more of a coalition leader who depended on the good opinion of other major landholders.

However, political alternative to the weak Shogunate was not the Emperor (or Emperors, during the period of the Northern and Southern courts), but the Shogun-appointed governors (Shugo) who held more and more authority over the shōen they were charged with managing, up to the point where shōen basically stopped existing.

This was a problem for the nobility ("nobility" here meaning the aristocrats who were associated with the Imperial court, not the Shogunate), since their revenue mostly came from shōen. So the Emperor's power base, already politically weak (because the Kamakura Shogunate had already weakened or negated their military power by tying military allegiance to their own land-granting authorities), now became economically weak as well. The dual system of having both Imperal- and Shogunate-appointed governors disappeared (possibly the Emperor still appointed people but they had no authority).

The Shugo supported the Ashikaga Shoguns because the Shoguns kept granting them more land and more authority over that land. The Ashikaga had no way of taking back that authority once granted, and so their authority (and central authority in general) eroded.

The Shugo continued to exercise more and more power over their domains, the most powerful known as Shugo-Daimyo and later just Daimyo (literally "great name," at first an informal distinction and later an actual office). The Ashikaga Shoguns' power continued to weaken until it collapsed entirely in the Onin War, which was a conflict between two Shugo-Daimyo. Then came the Sengoku period and fortunately another commenter covers what happened after that, since I'm less knowledgeable about the politics of that period. However by this point the Emperors had been deprived of almost all legal authority (by the Fujiwara and Kamakura), military authority (Kamakura), the ability to appoint officials with any real power (Ashikaga), and economic power (Kamakura and Ashikaga).

Sources: Japan, a Documentary History, David Lu

Japanese Culture, Paul Varley

The Tale of Heike, Helen McCullough (mostly the forward)

The court bureaucracy did not extend control outside Kyoto since the 10th century (if I remember correctly). Instead power was delegated to samurai of the provinces, appointed Shugo (protector) who raised their own forces to keep order in the provinces while Kyoto disbanded its own army. This would eventually lead to the Genpei War between two major samurai families and their allies and start of the Shogunates.

By then the only thing the emperor had left was legitimacy. So no real check on the emperor was needed as he did not have the resources to overthrow the shogun. What was needed was a check on other samurai families to prevent them from being strong enough to launch a rebellion using the emperor as an excuse. The way the Tokugawa did it (such as the sankin koutai arrangement) is outlined by /u/solarish below.

We should note that as hard as they tried, the emperor (or rather his position and legitimacy) did cause trouble when the Shogunates were weak. The Genpei War was fought in name over imperial succession. The actions of Emperor Go-Daigo basically brought down the Kamakura Shogunate and also challenged the new Ashikaga Shogunate forming the conflict of the Northern and Southern Dynasties. And of course the Bakumatsu when the Shogun Tokugawa Iemochi married the sister of Emperor Komei to try to take the pressure off the Shogunate from the court and its supporters. It didn't work as his successor Yoshinobu was overthrown when Satsuma-Choushuu-Tosa rose in rebellion and convinced Emperor Meiji to side with them.