The treaty of Tordesillas granted no rights of colonisation to the French and with the rivalry that grew between the powers of France under Francis I and Hapsburg dominions under Charles V, France attempted to explore and colonise where it could. The French king is even quoted as saying of the treaty "Show me the will of Adam!".
French exploration began in 1530 with Jacques Cartier sailing to OTL Hokkaido, Japan and the Kuril Islands. These he proclaimed as Francia Novia and claimed for Francis I.
He established Francesca on his first voyage and La Perouse on his second.
On his second voyage he sailed up into the mouth of the Amur River, which he named the St Lawrence River, as it was the feast of the St Lawrence, 1535.
Other settlements at Brest Nouveau and Plaisance followed by 1550.
Whaling and fur trade prospects made the lands of interest to the French.
The colony managed to establish mostly positive relations with the native Ainu, who were currently more angered by the Japanese incursions in the south of Hokkaido.
In 1544 the French also spread north, taking over the ailing colony of Henricia, long abandoned by most English colonists for lack of funding and supplies.
Exploration of the St Lawrence
After predominantly Basque fisherman had exploited the rich waters in the Sea of Angouleme and the estuary around the St Lawrence (OTL the Sea of Okhotsk and the Amur River), French fishermen and settlers soon followed. Interested in fishing stocks, lumber and furs, they followed the course of the river, eventually reaching Lake Baikal by 1564.
As Chinese influence on the Amur had dwindled since the Yongle era, French outposts and forts were erected along the Amur river basin, consolidating their rights to the resources there.
Exploration of the Yenisei
Thomas Aubert led an expedition from Fort Charles, on the east bank of Lake Baikal, down the Yenisei river in 1589. In two expeditions over the next year, they eventually reached the Arctic mouth of the river in the summer of 1590. However settlement did not occur in great numbers until the 17th century, with a lack of infrastructure and incentives in the frozen north, other than furs and lumber.