Yes, the expedition did know of Russia's presence, and so did the wider world. While the explorations of Russian pioneers like Chirikov and Bering were known among select audiences in Europe as early as the late 1740s, knowledge of Russian explorations was not initially disseminated widely due to the language gap and the generally lower level of discourse between Russia and other European nations. Russia and Spain, as the only two nations with firm presences in the Pacific Northwest until the 1770s, also had an interest in not widely publicizing their work. The British, as the insurgent power, had no such compunctions.

It wasn't until the accounts of James Cook's third voyage were published that knowledge of Russian possessions in the Americas became widespread, particularly in the English-speaking world. By the 1780s, and particularly after the Nootka Sound controversy of 1790, the English-speaking world was well-informed about circumstances in the Pacific Northwest.

In the 1780s, before even the Louisiana Purchase, and while he was serving in Paris as the American minister to France, Thomas Jefferson had extensive conversations with John Ledyard, a Connecticut man who served on Cook's third voyage.

Ledyard was the first American to circumnavigate the world (Cook's voyage began shortly after the Declaration of Independence, but Ledyard had not yet learned of it before the voyage began), and he was a strong advocate of the notion that the United States should explore the Pacific Coast because, as many Americans believed at the time, the United States was destined to span the entirety of North America and possibly South America as well.

"I wish to be on the coast before them (the British)," Ledyard wrote in 1785, "for they are the worst people in the world to follow in commerce or colonization among an uncivilized people."

Ledyard was foiled in his attempts to charter a ship for a mission from France to the Northwest, but in early 1786, Jefferson suggested an alternative plan: "I suggested to him the enterprise of exploring the Western part of our continent by passing thro St. Petersburg to Kamschatka, and procuring a passage thence in some of the Russian vessels to Nootka sound, whence he might make his way across the Continent to America."

Ledyard took up that idea, and he traveled across all of Eurasia, reaching Okhotsk, Russia's principal Pacific port, in 1787. During his travels, however, Russian officials became suspicious and believed that Ledyard was a spy. With the approval of Catherine the Great, Ledyard was arrested, declared persona non grata and deported from Russia via Poland. By 1789, he was exploring the Nile and drafting plans to explore North America from the east (via Kentucky) when he died of a disease caught in Egypt.

James Zug's American Traveler: The First Bicoastal American American Traveler: The Life and Adventures of John Ledyard, the Man Who Dreamed of Walking The World is the fourth and most recent biography of Ledyard, and I'd encourage you to pick it up if you get a chance. It's a fun read, as Ledyard led an incredibly fascinating life. I've used some of it to write this reply.

Despite Ledyard's struggles, American interest in the Pacific Northwest continued. The Cook missions had proved that there was enormous money to be made in the sea otter pelt trade with China, and in 1787, John Kendrick's Columbia Rediviva and Robert Gray's Lady Washington left Boston for the Pacific Northwest to participate in that trade. Both ships and both captains became involved in the Nootka Sound controversy, but both also became marked in history for their accomplishments on the voyage. Kendrick returned to Boston in 1790, having completed the first American-flagged circumnavigation of the Earth. Robert Gray, meanwhile, made the first confirmed discovery of the Columbia River, giving the United States a firm territorial claim on the Pacific Northwest by right of discovery.

At the same time the two were in the Pacific Northwest, the U.S. War Department attempted to coordinate exploration parties to the West, but these fell apart for various reasons. In 1790, a mission coordinated by a Lt. John Armstrong fell apart because of the difficulty of the challenge, and a subsequent mission led by a French naturalist was discontinued because the naturalist fell afoul of Federalist reaction to the French Revolution.

From 1795 to 1797, James Mackay and John Thomas Evans explored up the Missouri River, drafting maps and documents that would later be used by Lewis and Clark on their own expedition. I suggest Wood's Prologue to Lewis and Clark: The Mackay and Evans Expedition if you're interested in this "Lewis and Clark prehistory." At the same time, Alexander Mackenzie was traversing North America from Montreal, reaching the Pacific Ocean in 1793. His journals were published in 1801, and they provided another incentive for the United States to conduct an expedition.

When Jefferson was elected president and approved the Louisiana Purchase, he then had the means, motive and interest to put together an expedition that would have both the strength and the resources to travel across North America, then return. That expedition did not travel without predecessors, and it did not travel into a vacuum.