I'm particularly fond of Kona and believe that there's not really a good or bad time to visit - any time is great. do you want to stay in Kona or out of the city, like say at one of the resorts, such as Hilton Village, Royal Waikoloan, Mauna Kea. There are also a variety of places to stay in Kona, ranging from condos to hotels. My advice would be to check out a copy of "Hawai'i - the Big Island Revealed" for some of their suggestions.
You will find Kona hotter during June/July and wetter in Dec/Jan. It doesn't rain that often in Kona and usually when it doe, the showers are brief and warm. Most of the rain stays on the windward side (Hilo) or up in the mountains.
Make sure you have dinner at least one night at the Kona Inn - it has the best fish in Kona!