Although the Spanish had claimed all the coast from Mexico to the Bering Straits, it was Russians who were the first Europeans to actively explore and trade around the Alaskan mainland and nearby islands. According to the Historical Dictionary of the Discovery and Exploration of the Northwest Coast of America, in 1761 the fur trader Gavriil Pushkarev was the first Russian to set foot on mainland Alaska, specifically on the Alaska peninsula near the Shumagin Islands. In late 1761, Pushkarev, on an expedition financed by the merchant Ivan Bechevin,
reached the Alaska peninsula without knowing he had discovered
continental America and, trading with natives, prepared to winter over
at Protassov Bay opposite the Shumagin Islands.
As with so many European incursions into other lands in the pursuit of profit (in this case, furs), an encounter with the local Aleut soon led to fighting (in January 1762).
Map source: BioOne
Pushkarev is also mentioned in Alaska Peninsula National Wildlife Refuge (U.S. Fish and Wildlife Service) and in Brian Bonhomme's Russian Exploration, from Siberia to Space: A History which says:
The first Russian to land on the mainland of Alaska, rather than on an
island, was Gavriil Pushkarev, who stepped on to the peninsula in
Two years later, in 1763, Nikolai Daukin visited the Seward peninsula. Although the Spanish had been the first Europeans to reach the pacific coast of the Americas 250 years earlier (in 1513), it wasn't until 1775 that they ventured as far north as Alaska.
NOTE: in Alaska History: an Annotated Bibliography (2005), the chronology states "1731 Mikhail Gwosdev lands on the Seward Peninsula". However, other sources indicate that Gwosdev did not visit the area until 1732, and he was prevented from going ashore on the mainland by adverse conditions. However, he and the navigator Ivan Fedorov were the first Europeans to see Alaska (in 1732).