Hey, great question. The period that you'll be most interested in extends from 1558 to 1819 and from the Ural Mountains to what is now British Columbia in North America. During this time, Russia expanded across Siberia to the shores of the Pacific Ocean. Faced with an immense ocean obstacle, the Russians drove themselves onward, crossing that ocean and establishing themselves upon what is now known as Alaska. They drove still further, establishing trading posts in California and Hawaii.

Throughout these 300 years, one thing principally drove Russian expansion: furs.

As the Stroganov Chronicle states:

"Then through God's grace the Muscovites and the cossacks ... became very wealthy from trading in furs, and they lived in joy and happiness, thanking Almighty God for having given the Sovereign such rich land."

It was fur wealth that encouraged the eastward advance. Furs were valued in China, and Chinese goods were valued in Europe. The Treaty of Nerchinsk in 1689 settled the Russian-Chinese boundary to a large degree, and it also created the trading city of Kiakhta, located directly on the Russian/Chinese border in the middle of Siberia. It was here that the Russians carried furs, and from here that they carried Chinese goods on an immense transcontinental journey to Europe.

Enormous amounts of money were made from furs, and this led to overhunting, which in turn led Russians to constantly look eastward and eventually to North America. 1558 is the year that Tsar Ivan Vasilevich granted Grigorii Stroganov financial, judicial and trade privileges along the Kama River to the east of Kazan.

This claim brought the Stroganovs into conflict with Kuchum Khan, a descendant of the Mongol Empire who held a shaky dominion over a large portion of Siberia. In 1581, the Stroganovs funded a band of cossacks led by Ermak Timofeev. Ermak's cossacks crossed the Urals and conquered huge swaths of land, and even though Ermak and most of his party were killed, Stroganov reinforcements held much of the terrain they took and implemented a procedure of conquest and tribute that would extend to North America.

Even today, Siberia is lightly settled. Between 1581 and the end of the 18th century, this was true even more so. Instead of outright control of a particular piece of (otherwise valueless) terrain, Russian authorities set up a system of tribute called the yasak.

After the Stroganovs' success (and huge riches earned from tribute), the Russian government authorized other ventures across Asia. These operated much as the Stroganov venture did. A well-armed force would advance and defeat the armed forces of a tribe, khan or region. In turn, that defeated group would swear loyalty to the tsar and agree to pay a tribute to the conquering agency. That tribute might be paid in furs, gold, or other precious, tradable objects. In exchange, the armed force would agree to not kill (pretty much) everyone and restrict itself to petty corruption, extortion and massacre. If the group continued to resist, the Russians would fight them and convince the group's neighbors to pile on by pledging a share of the loot.

A beaten group might be forced to supply corvee labor for Imperial projects, or be forbidden to conduct diplomacy. It might be required to trade only with a particular trading company or trader. It would be required to follow the Orthodox church, discard other religions and give Orthodox priests and monks safe passage.

The Russians built a series of small forts ostrozheks and large forts ostrogs at strategic locations, not unlike what the United States would do in the 19th century. State employees and colonial officials would oversee this forcible tribute traffic and ensure the imperial government got its proper take. The streltsy (musketeers) provided firepower to back up the cossacks who did much of the conquering.

These techniques extended across the Pacific Ocean to Alaska during much of the 18th century. There, the prime fur was sea otter, but the Russians' inadequate naval skills meant they had to compete against the British and Americans for the fur trade in North America. Furthermore, the Russians were prohibited from trading at China's Pacific ports, while the other European powers were allowed.

The march of progress and the Enlightenment also put pressure on Russia to reform its practices. Catherine the Great formally forbade the practice of yasak, and Tsar Paul granted a monopoly in North America to the Russian-American Company in order to prevent some of the worst competition-driven offenses there.

It was not until the RAC's second charter was issued in 1819, and the company's American operations came under the authority of a naval officer, that the worst parts of the yasak came to an end. About the same time, the sea otter trade collapsed under the weight of overhunting, and the Russian government instituted some of the world's first environmental protection measures to allow the continued, sustainable harvest of Alaska fur seals.

The study of the Russian advance got a lot of fuel in the 1970s and 1980s, during a period of detente between the Soviet Union and the United States. The Oregon Historical Society in particular deserves a great deal of credit for translating some 3,000 pages' worth of Russian documents into English in the last five years of the 1980s. These documents were published in a three-volume set, and they're an incredible English-language primary source with the subtitle To Siberia and Russian America. There are plenty of popular works about the Russian conquest of Siberia, which was completed with the 1858 Treaty of Aigun.

The response by The_Alaskan below is an excellent one, especially on the issue of Russian expansion into Siberia. I think his response with regards to the expansion across the Pacific glosses over some important details. He refers to the yasak process, without really going into details of how it affected the native population in the Aleutians, the Bering Sea, and the remainder of Alaska (Russian America). In fact, starting with the first fur-gathering expeditions (e.g. Chuprov's Eudoxia voyage- 1745), the relationships with the natives were marked not only by misunderstanding, but by massacre, war, and slavery. The first populated island Chuprov landed on (Agattu) was the site of the first “misunderstanding”. Records are unclear, but it seems that several natives were shot after a dispute. On the next island (Attu), the Russians tried to trade, and demanded that the Aleuts give hostages for good behavior. Disputes over relations with native women led to the killing of at least 15 natives in one village. Settling down for the winter, the Russians took control of the native women in the villages they “settled”, and forced the males to hunt for them, occasionally with pay, but normally without. Threats were made to destroy the villages and food stores, if the hunters did not bring back enough furs (and the reality of these threats was demonstrated).

Though there were some “good” Russians (e.g. Tolstykh), the majority of the fur traders in subsequent years followed the example of Chuprov.

The Yasak process (taxation based on the concept of tribute and swearing allegiance to a foreign lord) was not formally imposed on the Pacific area until 1748 by Tsarina Elizabeth, and was frequently ignored or used to essentially steal anything the natives had-- it was not administered fairly or for the benefit of the government. The government took some weak initiatives to protect the natives from Alcohol and venereal disease, but without much effect. The Aleuts saw little benefit from the Yasak (which was supposed to provide protection from the state, among other benefits), and resisted paying the taxes. One Yasak collector (Pushkarev) in retaliation reportedly killed 23 Aleut girls and committed many other crimes.

As a result, Aleuts on Unalaska revolted in the early 1760s, and destroyed 4 ships with a large number of their crews. In retaliation, 18 Aleut villages were totally destroyed on Unalaska, as well as numerous others on surrounding islands. Aleut property and boats were burned, and torture was used to find natives in hiding. A later Russian in Alaska (Bishop Veniaminov) wrote that one time Soloviev had a dozen Aleuts tied in a row to see how many bodies it took to stop a musketball (the answer was reported as 9).

Tsarina Catherine supported the fur traders in the Pacific through a “laissez faire” policy, which while it saved the government the problems and costs of control, still failed to stop the maltreatment of the natives. She issued a ukase demanding kind dealings with the natives, which most of the traders ignored. Shelikov was maybe the exemplar of a trader who tried to maintain good relations with the natives-- He formed probably the first large-scale fur trading company in Alaskan waters, about 1783. He formed a settlement and trading post at Three Saints Bay on Kodiak Island in 1784, and by all reports kept his personnel under tight control. He developed good relations with the Kodiak natives, and paid them for work done.

The government in Moscow seems to have wanted to stop some of these widespread depredations, and that goal was partially the impetus for the later formation of the Russian American Company (RAC). Early years of the company, however, continued the forced labor of the Aleut males in the search for furs. As years progressed, Many Aleuts worked voluntarily for the RAC, and even fought alongside the RAC in combat with other tribes (e.g. the Tlingits around Sitka).

Under the RAC, depredations continued (though at a lower level), and open warfare occurred occasionally, as at Yakutat and Sitka. Against this must be placed the actions of the RAC in readily accepting children of mixed (Russian and Indian) ancestry to schools and into the employ of the RAC (including sending some to Russia for technical training). Many accusations of mistreatment of the natives are well-documented in documents provided by representatives of the Russian Orthodox Church who lived and preached in Alaska. However, these latter accusations need to be understood in the context of the political in-fighting which raged for decades between the Church and the RAC. They probably did occur occasionally, though were probably exaggerated by Church reporters including Bishop Venimeniof. The RAC, though far from a perfect employer or entrepreneur, seems to have eliminated the worst of the transgressions which were occurring prior to its formation.

This website is too short to allow a full description of the subject, but there are numerous works available, including Lydia T. Black, “Russians in America, 1732-1867”, and Hector Chevigny “ Russian America/The Great Alaskan Adventure 1741-1867”.

It would help if you clarified your question. Are you looking for information about Russian interaction with natives as they moved across Siberia to the Kamchatka Peninsula, or about Russian interaction with natives after the Russians began to spread East from Kamchatka? Most information about the former (which is what I suspect you are asking about) is published in Russian, and is not my area of interest. There is lots published in English on the Russian spread into the Pacific and into Alaska.