Map from Google - Bangladesh

The climate of Bangladesh is tropical, with a pleasantly warm and sunny winter from November to February, a short hot spring between March and May, and a long rainy season from June to October. The country is flat and crossed by a thousand branches of the huge delta of the Ganges-Brahmaputra, and is therefore exposed to floods, as well as rogue waves and high tides when cyclones hit the Bay of Bengal.

The dry season runs from November to March. The monsoon has left the country during the month of October, but sometimes the rains can last until November, especially in the south-east, or when a late cyclone arrives. In general, however, November is hot, but not too much, with highs around 85 °F, and sunny.
The real winter begins in December, and is characterized by warm sunny days, followed by cool nights. In the far north, in cities like Saidpur and Rangpur, lows in January, the coldest month, are around 51 °F, and highs around 73 °F.
Here are the average temperatures of Saidpur.
Average temperatures - Saidpur
Min (°F)525563707577797977736455
Max (°F)737988919090909088888477

In the central area, which includes the capital Dhaka, the temperatures are higher, around 55 °F at night, and 78 °F during the day.
Average temperatures - Dhaka
Min (°F)556168757779797979756657
Max (°F)778290939190889090908679

Along the southern coast, the night minima are slightly higher, because of the influence of the sea.
During winter, from December to February, sometimes the north wind can bring some cold at night, with lows around 37/43 °F in the north, and around 50 °F in the south-central and in Dhaka.
In February the temperature begins to rise, and from March it begins to get really hot. Spring, from March to May, is actually the hottest season: the average maximum temperatures reach 95 °F in April in inland areas, while they don't go above 90/91 °F along the coasts, where the hottest month, for the natural thermal inertia of the ocean, is May. The highest peaks are about 108/109 °F in the north, 104 °F in Dhaka and in the center, and 99/100 °F in the south. The relative humidity gradually increases, and the heat becomes more stifling; in March the first heat thunderstorms occur, called northwesters, with hail and wind gusts, so that in May more than 10 inches of rain fall in the south and 6 inches in the north.

The real monsoon arrives from late May to early June, starting from the south-east. It brings a more compact cloudiness, high humidity even during the day, frequent rains, but also a decrease in temperature, which drops to 86/90 °F during the day, but remains high, at about 77 °F, at night. The rains are more abundant along the south coast, particularly the south-east (see Chittagong, Cox's Bazar), but also in the north-east, close to the mountains of India (see Sylhet). In Sylhet, in July 31.5 in of rain fall, 29.5 inches in Chittagong, 35.5 inches in Cox's Bazar, and even 40 inches in Teknaf.
The monsoon is less intense in the west, where the annual quantity of rain is around 60/63 in, and in July it reaches about 12/14 in, as happens for example in Pabna, Jessore and Rajshahi. In Dhaka, precipitation amounts to 85 inches per year, of which 14.7 inches in July. Here is the average precipitation in Dhaka.
Average precipitation - Dhaka
Prec. (in)0.312.66.113.413.214.812.412.

In Chittagong, on the east coast, the rains reach 115 inches per year. Here is the average precipitation.
Average precipitation - Chittagong
Prec. (in)

More to the south, rainfall reaches 138 inches per year in Cox's Bazar, and up to 158 inches in the more southern Teknaf. In Sylhet, in the northeast, where as we said the other very rainy area is found, they reach 158 inches per year.
The monsoon gradually withdraws between October and November.

Bangladesh is particularly vulnerable to tropical cyclones, which come from the Bay of Bengal, usually from late April to early November (but sometimes they also occurred in late November and early December). In general, the periods in which they are most dangerous are both the beginning and the end of the monsoon period, that is, May-June and October-November. The most destructive cyclone on record occurred in October-November 1876, in November 1970, and at the end of April 1991.

The sea is warm enough for swimming all year round, although it drops to 75 °F in winter, as we can see in the following table.
Sea temperature - Cox's Bazar
Cox's BazarJanFebMarAprMayJunJulAugSepOctNovDec
Sea (°F)757579828486848484848277

When to go

The best time to visit Bangladesh is winter, between December and February. Typically in November the weather is good, and it's hot but not too much, but we cannot exclude that some cloud banks could linger on, or even some late cyclones. In March and April it gets very hot, and thunderstorms in the afternoon or evening, sometimes intense, are possible.

What to pack

In winter: spring/autumn clothes, light for the day, a jacket and a sweater for the evening, possibly hat and scarf for cooler evenings in the north (see Rangpur and Saidpur).
In summer: tropics-friendly, lightweight clothing of natural fibers, light raincoat (if you can stand the heat) or umbrella (assuming that it is effective against the monsoon rains).