Late spring (May, June) and early autumn (September, October) are the best windows — long days, comfortable temperatures, and the local wineries open for visits without the August crowds. The vineyards turn yellow and red in mid-October, which is the prettiest time of year in the surrounding country.
The Marché Médiéval in early August fills the old town with re-enactors in period dress, period food stalls, and theatrical performances inspired by Joan of Arc — Chinon's only really busy weekend of the year. Winter is quiet and cold, but the fortress and a couple of restaurants stay open, and you will often have the old town more or less to yourself.
The Thursday morning market in Place Jeanne d'Arc runs year-round.