Louis XIV (r. 1643–1715) didn't found* New France—that was Louis XIII (1634, founding of Trois-Rivières) and earlier explorers (Cartier, Champlain). But the Sun King did put the colony on a demographic footing—through a brilliant, weirdly specific policy.
Between 1663 and 1673, the Crown recruited ~800 young women (orphans, poor but virtuous), gave each a dowry (50–300 livres), a trousseau, and free passage to Quebec. They married quickly—within weeks—because the colony had a 6:1 male-to-female ratio. Result: a population explosion. By 1681, New France had ~10,000 Europeans (from ~3,000 in 1663).
Licensed (voyageurs) and unlicensed (coureurs des bois) fur traders penetrated the interior, married Indigenous women (à la façon du pays), and created the Métis nation. They weren't "sent" by Louis XIV—they were tolerated because they extended French commercial reach to the Great Lakes and Mississippi.
The Carignan-Salières Regiment (1,200 men, 1665) was sent to crush the Iroquois. ~400 stayed as settlers after the 1667 peace. Land grants (seigneuries) anchored them.
Louis XIV's genius: he solved the colony's demographic crisis with targeted, state-funded marriage migration. Cheap, effective, and very French.