In Kolkata (or Calcutta), the capital of the state of West Bengal, the climate is tropical, with a sunny and pleasantly warm winter from December to February, when, however, sometimes at night the temperature can drop below 10 °C (50 °F, so it's better to bring a sweater and a light jacket), a very hot period before the monsoon, from March to May, when the temperature can reach or exceed 40 °C (104 °F) in the worst moments, and a rainy season from June to mid-October, characterized by stifling weather, widespread cloudiness and frequent rains. From mid-October to mid-November, before winter, there is a period of one month characterized by good weather and high temperatures, around 30 °C (86 °F).
Here are the average temperatures.
Kolkata - Average temperatures
Min (°C)141721252627262626241914
Max (°C)262934363634323232322926
Min (°F)576370777981797979756657
Max (°F)798493979793909090908479

The monsoon season goes roughly from June 5 to October 15, but already in May (and more rarely in April), the first showers in the afternoon may occur. The total annual rainfall is 1,600 millimeters (63 inches). From November to April, the sun shines and it almost never rains.
Here is the average precipitation.
Kolkata - Average precipitation

The sun in Calcutta often shines in the dry season, while in the rainy season the sky is often cloudy.
Kolkata - Sunshine

From mid-April to early December, Calcutta can be affected by cyclones, tropical storms coming from the Bay of Bengal, with a higher chance at the beginning and at the end of the period, namely from April to June, and in October and November. The Bengal region is particularly prone to flooding because of the conformation of the territory (we are in the immense delta of the Ganges-Brahmaputra, and the river of the city, the Hooghly, is a branch of the Ganges).

Victoria Memorial Hall

South of Calcutta, the sea is warm enough for swimming throughout the year, although it drops to 23/24 °C (73/75 °F) in January and February.
Kolkata - Sea temperature
Temp (°C)232426282930292929292725
Temp (°F)737579828486848484848177

Best Time

The best time to visit Kolkata is from mid-December to February. In November and in the first half of December, the rains have usually ceased, but every now and then a cyclone can still affect the area. In March, it starts to get very hot.