Wikipedia is actually quite useful for that. It has a list of airports in the country. I can easily find flights to some of them on the usual search engines, mostly with TAP through Lisbon. TACV, the flag carrier, also appears to offer many flights.
If you want to do a “broad search”, without specifying an airport, you can also use rome2rio, Skyscanner or Adioso. All three accept a country name as destination and the first one even searches other modes of transportation and some charter/low-cost carriers.
If you don't find what you want through the search engines, you can also check the airport pages on Wikipedia. There is an article about most of them, typically with a mostly up-to-date list of destinations and carriers. Thus, you can for example easily find out that TUIFlyNordic offers flights to Aristides Pereira International Airport from Arlanda (apparently sold through Fritidsresor.se).