Chennai exhibits a moderate hot temperature throughout the day with moderate humidity as it is along the coastal lines of the Bay of Bengal.
March to May form the summer and June to August you can find monsoon season but dependant on the rain from Western Ghats. If the monsoon fails then the Monsoon rain on coromondal coast will be during September to November.
Average temperature during summer would be around 40°C and to the max it would go up to 42°C. During Winter season average temperature will be around 25°C during day time and the night temperature will reduce up to 20°C.
You can travel to Chennai any time throughout the year. City will be generally hot except between months of December to February where in you can find some rainy days with mild chillness during the day time. If you really want to avoid summer season and the hot temperature then you can try going between the months of November to February. Temperature would range from 25°C to 30°C during these months.