To find your Connecticut ancestor's newspaper obituary, you have to start by finding the correct newspaper. Sadly, this isn't easy as you might hope. Connecticut only has a few major newspapers now, all of which cover slightly different parts of the state. That wasn't true twenty, seventy-five or two hundred years ago. The number of newspapers and their coverage region has changed dramatically overtime.
While there's no ready source for determining a newspaper's coverage area, the Library of Congress has a resource that makes identifying the local newspapers much easier. The Chronicling America Online Newspaper Directory is an online index of the newspapers published for an area. You can narrow it by time period. You will still miss some newspapers - Chester obituaries often appeared in the Middletown Press - but it's a starting point.