As noted, you need a better economy, since its currently subsidised by Denmark. Not to mention that Greenland has serious social issues which are similar to Inuit communities in the US and Canada--alcoholism, domestic violence, and suicide (by far the highest rate in the world) are common.
Maybe get some sort of government-sponsered settlement program in the early 20th century. Denmark sends in a few colonies of farmers, like was done in parts of Alaska. Granted, they'd be pretty reliant on government subsidies like the rest of Greenland, but the point is there's now a larger population. Most will likely either move to Godthåb, back to Denmark, or maybe to the US or Canada before long. Maybe an early 20th century craze for
roseroot, an Arctic/alpine plant used in traditional medicine which is said to be a mood booster would give a better crop than moving to Greenland to farm potatoes or whatever. At the same time, Denmark spends more money on their navy for the sake of their global empire (so no selling the Virgin Islands TTL), and puts a base in Greenland. Later on NATO uses this base, plus also the one in Thule. Hydroelectric power and related industries (i.e. aluminum smelters) aree developed in Greenland earlier as well.
Sometime in the early 21st century, a tightly divided referendum is held on Greenlandic independence, which narrowly passes. Being that Greenland TTL has more population, and most of them are Danes (although the better economy in Greenland means fewer Greenlanders move to Denmark), Greenland is more multiethnic than being an Inuit state--both Danish and Inuit are the native languages. The government uses much of the revenue from hydro power and especially the recent mining projects to combat the social issues amongst native Greenlanders.