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 30 °C, 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 11 °C, and highs around 23 °C.
Here are the average temperatures of Saidpur.
Average temperatures - Saidpur
SaidpurJanFebMarAprMayJunJulAugSepOctNovDec
SaidpurJFMAMJJASOND
Min (°C)111317212425262625231813
Max (°C)232631333232323231312925

In the central area, which includes the capital Dhaka, the temperatures are higher, around 13 °C at night, and 25 °C during the day.
Average temperatures - Dhaka
DhakaJanFebMarAprMayJunJulAugSepOctNovDec
DhakaJFMAMJJASOND
Min (°C)131620242526262626241914
Max (°C)252832343332313232323026

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 3/6 °C in the north, and around 10 °C 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 35 °C in April in inland areas, while they don't go above 32/33 °C along the coasts, where the hottest month, for the natural thermal inertia of the ocean, is May. The highest peaks are about 42/43 °C in the north, 40 °C in Dhaka and in the centre, and 37/38 °C 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 250 millimetres of rain fall in the south and 150 mm in the north.

Bangladesh
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 30/32 °C during the day, but remains high, at about 25 °C, 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 800 mm of rain fall, 750 mm in Chittagong, 900 mm in Cox's Bazar, and even 1,000 mm in Teknaf.
The monsoon is less intense in the west, where the annual quantity of rain is around 1,500/1,600 mm, and in July it reaches about 300/350 mm, as happens for example in Pabna, Jessore and Rajshahi. In Dhaka, precipitation amounts to 2,100 mm per year, of which 370 mm in July. Here is the average precipitation in Dhaka.
Average precipitation - Dhaka
DhakaJanFebMarAprMayJunJulAugSepOctNovDecYear
DhakaJFMAMJJASONDY
Prec. (mm)7256515534033537531531517535152155
Days13381518212116810115

In Chittagong, on the east coast, the rains reach 2,900 mm per year. Here is the average precipitation.
Average precipitation - Chittagong
ChittagongJanFebMarAprMayJunJulAugSepOctNovDecYear
ChittagongJFMAMJJASONDY
Prec. (mm)5255514030560073554026519070152940
Days02371218222016930112

More to the south, rainfall reaches 3,500 mm per year in Cox's Bazar, and up to 4,000 mm in the more southern Teknaf. In Sylhet, in the northeast, where as we said the other very rainy area is found, they reach 4,000 mm 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 24 °C in winter, as we can see in the following table.
Sea temperature - Cox's Bazar
Cox's BazarJanFebMarAprMayJunJulAugSepOctNovDec
Cox's BazarJFMAMJJASOND
Sea (°C)242426282930292929292825

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 fibres, light raincoat (if you can stand the heat) or umbrella (assuming that it is effective against the monsoon rains).