May and September are the best months. Warm but not hot, fewer cruise ships, swimmable sea, every restaurant and museum open. The walk up to the fortress is sweat-free in May and possible in September; in July and August it can be punishing.
June and early October are pleasant shoulder season — warmer water in early October than in June, slightly more reliable weather in June.
July and August are hot (32–35°C in the shade), busy with cruise ships, and exhausting. Hotels triple their rates. If you do come in peak season, time visits to the old town for very early morning or evening when the cruise crowds have left.
Carnival in mid-February is the most distinctive local event — a costumed parade through the old town, with masked revelry that draws on a Venetian tradition the Boka has kept alive. St Tryphon's Day (3 February) brings the procession of the Bokeljska Mornarica through the cathedral square.
Winter is mostly closed: many restaurants and museums shut, but the old town is unrecognisably empty and the surrounding mountains can have snow on them down to about 1,000 metres. Atmospheric for a short visit.