November to February is the best time to visit Chennai, with December and January the strongest months.
Temperatures in this window are comfortable — generally in the 20s during the day, cool in the mornings and evenings — and rainfall has largely tapered off after the November peak. This is the only period of the year when you can walk around Fort St. George or along Marina Beach in the middle of the day without struggling.
December is exceptional for a specific reason: the Margazhi music season, when Chennai hosts one of the world's largest classical music and dance festivals across dozens of sabhas (concert halls). Hundreds of performances run across roughly six weeks, many of them free or very cheap. If you have any interest in Indian classical arts, this is the time to come. Book accommodation early, as the season draws visitors from across the world.
Mid-January brings Pongal, Tamil Nadu's harvest festival and its most important celebration — a good time to see the city at its most festive, though many businesses close for several days.
Avoid March to June unless you have a specific reason. The heat is punishing and it materially limits what you can do during daylight hours.
October and November bring the heaviest rain. Travel is possible but flooding can disrupt plans at short notice.