Chennai is hot and humid year-round, but the differences between seasons are meaningful.
November to February is the best period for visiting. Temperatures are in the mid-20s Celsius, humidity is lower, and the monsoon has passed. This is the Margazhi season — the Tamil music and dance festival month of December–January, when the city hosts hundreds of classical Carnatic music and Bharatanatyam dance performances. If you're interested in classical South Indian arts, visiting in December is specifically worthwhile.
March to May is hot and increasingly humid, with temperatures reaching 40°C and above in April and May. Manageable with air conditioning, but uncomfortable for extensive outdoor sightseeing.
June to September brings the southwest monsoon, which hits Chennai less directly than the western coast. Rain is intermittent rather than constant. October and November see the northeast monsoon, which is Chennai's main rainy season — heavy rain is possible, and flooding in low-lying areas is common.