Chile can be visited all-year long, mainly because of its wide range of climates (from desert to toundra, with Mediterranean and subtropical climates as well). Based on my own experience, this is what I would recommend for the main areas of the country.
Atacama desert: all-year long, best from April to November
The climate in the North of Chile does not change much. From La Serena to the border with Peru, you will find dry, temperate weather during the day, all year long. It can get pretty hot during Summer as the sun is very strong, but it never gets rainy or cold (unless you reach very high altitudes in the Andes). If you plan to combine your trip to Atacama with Bolivia or southeastern Peru (i.e. Machu Picchu and Titicaca lake), you should definitely avoid January-March, as it is the raining season there.
Central Chile: from October to early May, best in November, December, March and April
The central area of Chile, which includes Santiago, Valparaiso, Viña del Mar, and the Wine country, benefits from a pleasant Mediterranean climate. Which means it never gets too cold, neither too hot, nor too rainy. Most Chileans and Argentinians take their holidays in January-February, so you may want to avoid going to the coast at this time of the year. It is nothing compared to the beaches of Spain or the south of France, though. But if you look for quietness, November, March or April are much better.
Lake & Volcanoes districts, Chiloe: from November to early April
The regions of the lakes and volcanoes, as well as the Chiloe islands, are known for their heavy rains. The weather is usually better in Summer. If you intend to do water sports such as kayak, rafting or even fishing, consider that the snow is still melting in November-December, therefore the currents in the rivers will be much stronger.
Patagonia and Tierra del Fuego: from November to early April
While the Summer rarely gets warm in Patagonia, it surprisingly never suffers from very cold temperatures. While January-February are more sunny, the wind is very strong at that time of the year. November-December, March-April can be wetter and cooler, but less windy, which is a good compromise with hikers in Torres del Paine.
Easter Island: All-year long, best in April-May and October-November
Easter Island benefits from a subtropical climate. There are sporadic rains all year long, and the humidity can get suffocating in Summer. This is why it is better to avoid the period from December to March.
Winter sports: Late June to Late September
While Chile has the highest mountains in America, the climate does not allow a long winter season. The area around Santiago is the best equiped with several ski resorts, however, there is usually more snow in the south, at Termas de Chillan and Pucon, among others. The winter season is gaining momentum in Patagonia, as the weather is not too cold and not too windy there, and snowy landscapes are beautiful. But you need proper equipment and preparation in order to go to Patagonia in Winter.
Chile overall: November-December or March-April
If you plan to see a bit of Chile from North to South, the months starting and finishing Summer are probably your best bet. If you come in November, start with the North or the Center of Chile, and then head south. If you come in April, start with Patagonia and finish in the desert or Easter Island.