There does not seem to be any regular, scheduled and affordable direct service between North America and Greenland.
Rome2rio only finds flights via Iceland.
Wikipedia states:
There are no direct flights to the United States or Canada, although there have been flights Kangerlussuaq – Baltimore, and Nuuk – Iqaluit, which were cancelled because of too few passengers and financial losses. An alternative between Greenland and the United States/Canada is Icelandair with a plane change in Iceland.
From the larger airports (I didn't check the smaller ones), Flightconnections only shows internal flights, or flights to Iceland or Denmark.
There are apparently no ferries either.
The Royal Arctic Line, which operate cargo ships (there's sometimes a way to hitch a passenger ride on some cargo ships) only have routes to Iceland and Denmark (or elsewhere in Europe).
Swoop Arctic state:
Whether travelling by air or sea you’ll arrive from the east as there's no access from North America.
and
You can only fly to Greenland from Copenhagen & Reykjavik, Keflavik & Akureyri in Iceland. There are currently no direct flights from North America.
and
Flying is the most popular route to Greenland as there’s no international ferry service.
But they mention:
join an expedition cruise from Svalbard or the Canadian Arctic.
They do have packages including such cruises, but prices are above the $10K mark.
Exploring the description of the cruise, they use the Ocean Endeavour, which is operated by Adventure Canada, which indeed have expedition cruises between Canada and Greenland (though not all in the right direction for you), but the price is still in the $5K+ range.
There may be options involving private jet repositioning flights/empty legs, but that would probably require a good deal of luck, and I'm not familiar with the best sites for this. And I have no idea what they would charge.
It may be worth it to follow arrivals/departures on Flightradar24 or Flightaware for some of the airports for a few days. One can see there are flights from Air Nunavut (mentioned in the comments to the question, but they seem to be charters so may be expensive) as well as Canadian North (don't see those routes on their website, not sure if those were charters or what). There are also quite a few private planes.