The season runs between May and Mid October. Many hotels shut, ferries stop running and there are no kayaking tours outside these months. 

May: Low 14 °C (night) High 21 °C (day); 8 hrs Sunshine; 75 mm rain; Sea 18 °C
May marks the beginning of the season for Dubrovnik and its surrounding Islands. The month is marked by typically modest rainfall and pleasant temperatures. The sea is still warming up. The sea is warming up, but is comfortable if swimming. Many high end hotels (particularly on the Islands) offer big discounts on usual high season rates.

June: Low 18 °C (night) High 25 °C (day); 10 hrs Sunshine; 48 mm rain; Sea 22 °C
June (along with September) is arguably the optimum months to visit. Rainfall is very low and temperatures are consistently in the high 20s without the uncomfortable heat that can visit in the height of summer.

July:  Low 21 °C (night) High 29 °C (day); 12 hrs Sunshine; 26 mm rain; Sea 24 °C
July marks the beginning of the peak season. Temperatures are high, there is lots of sunshine and the sea is warm. Many Dubrovnik hotel prices double during July and August. 

August: Low 21 °C (night) High 28 °C (day); 11 hrs Sunshine; 38 mm rain; Sea 25 °C
Weather as July. The heat is usually manageable, particularly out on the islands, and most tourist accommodation has air conditioning. There are occasional heat waves during July and August which can bring uncomfortably hot days (particularly if you are stuck in the City).

September: Low 18 °C (night) High 25 °C (day); 9 hrs Sunshine; 101 mm rain; Sea 23 °C
September (along with June) is an optimum month to visit Dubrovnik and the Elaphiti Islands. Temperatures are perfect, the sea is still warm (so great for snorkelling) and whilst it is still busy, the larger volumes of tourists are gone. There is usually more rain in September.

October: Low 14 °C (night) High 21 °C (day); 7 hrs Sunshine; 162 mm rain; Sea 21 °C
October can be a mixed month in the Med. It often throws up the best conditions with few people around, warm sunny conditions and a still comfortable sea temperature. However if you are unlucky, it could mean a week of rainy blustery conditions. As with May, most of the best hotels offer stays at a fraction of their high season rates.

Sources: BBC Weather, holiday-weather.com