Hey there, /u/DoubleDgit. I'm off work now and can answer your question.

In short, fur trading. I suggest Lydia Black's Russians in Alaska: 1732-1867 as the best book on the Russian period in Alaska.

During the 17th and 18th centuries, Russia expanded eastward through Siberia, setting up a system of fur-based tribute in which officials would gather furs from an area of Siberian Native villages. These officials were backed by the military, and in the middle 18th century, following the Russian discovery of Alaska, this practice moved into Alaska.

By the end of the 18th century, there were several competing Russian trading companies in Alaska. These companies employed native Aleut and Alutiiq hunters to catch sea otters, whose furs were incredibly valuable in China. A type of triangular trade developed, whereby furs would be shipped from Alaska to China for sale. In China, traders would buy Chinese goods (porcelain for one) that would then be taken back to European Russia for sale. Proceeds from the sale would buy supplies for the Alaska operations and pay dividends to the trading companies' shareholders.

This was an enormous undertaking operated by surprisingly few Russians. At no point in the Russian period (1732-1867) were there more than 1,000 Russians in Alaska.

Most of the work in Alaska was performed by Natives, and in the beginning, they were coerced to hunt furs through extraordinarily violent methods. Massacre, hostage-taking, and attacks on competitors were common. So bad was the situation that Catherine the Great felt that something had to be done to prevent the mistreatment of Natives.

She issued several orders calling for fair treatment, but the biggest success (at the hands of her successor, Emperor Paul) was the establishment of the Russian-American Company, a monopoly with trading power over all of Alaska. This company was the successor to one of the most successful trading companies operating from Russia's Far East.

Through three 20-year charters, the RAC operated with a single mission: Make money. There was no drive toward settlement, no effort to establish towns, as there was in the British and French territories. Russia had all of Siberia to settle, after all, and Alaska was first and foremost a money-making opportunity. There were only such settlements as were needed to support the fur trade.

As time went on, Russian America started to change. Intermarriage between Russians and Natives (plus Natives who Russianized) created a Creole class that willingly worked for the RAC, often to great success. Alaska Natives attended navigation colleges in European Russia, attended Russian churches and shopped in Russian stores.

The fur trade changed as well. The triangular trade turned square as Americans arrived on the Pacific Coast of North America. It was much cheaper to buy supplies (and they were much more reliable in arriving) from California instead of Russia proper. Before the Americans took over the Pacific Coast, the Russians even tried establishing an agricultural colony in California as a way to supply their holdings farther north. Called Fort Ross, this colony was never as successful as planned, and it was sold a few years before the California Gold Rush.

The Russians attempted to diversify their fur-trading operations, but were never successful. American and British whalers filled the waters south and west of Alaska, but the Russo-Finnish Whaling Company of the 1850s was a commercial failure. Gold, though known to exist in an unknown quantity, was not exploited, and neither was coal. In fact, the biggest commercial success was ice. Shipments of ice (starting in the 1850s) were undertaken regularly between the Kodiak archipelago and San Francisco.

Despite the distances involved, despite all the problems of a remote colony, the Russian American Company stayed largely profitable until the 1867 Alaska Purchase Treaty. Russia never had a big presence in Alaska, but it never needed one -- its mission was money, not territorial acquisition or settlement.

Like many in the colonial game control of land and profit were some of the primary reasons. To start, control of land was very desirable. The Tsar was pleased to look at a map of the Russian Empire and see how much of the world was his, land value aside. Since there were so many competitors in the colonial game, empires tried to claim land before someone else did. However, control also required boots on the ground and merely planting a flag did not always work. For the Russians, the Russian America Company was forced back by the indigenous Tlingit in Alaska until they resettled in an uneasy truce. Outside the forts, the Russians didn't really control Alaska.

Profit was incredibly important. Desire for profit was often the driving force behind empire. The RAC did well early on with the hunting of Alaskan sea otters, thanks to the massive market for otter pelts in China. However, they quickly over hunted and destroyed their source of profit and Russian America became a unprofitable and largely forgotten corner on the edge of Russia's massive land empire.

Other overseas Russian colonies were built for other reasons. Fort Ross in California was built to supply food to the Sitka settlement in Alaska. This failed and was sold to the enterprising Swiss con man, John Sutter.

One of the biggest drives for Russians was fur trading. It was very lucrative. However, not to be underestimated is the movement of religious groups. Old Believers were often found on the fringes of Russia's expanding empire in the 18th and 19th century. To this day there are Old Believer communities in Alaska. Penal colonies also played a role, not so much in the pushing of borders, but more in developing/settling territory. If you're looking for more information, I highly recommend W. Bruce Lincoln's "The Conquest of a Continent: Siberia and the Russians."

EDIT: Looks like I was beaten to the punch.