Kakadu has two very distinct dry and wet seasons. The dry season runs from May to September, and is generally considered to be the best time to visit the park. Most of the roads are open during this time, and it's the coolest time of the year. The park tends to get very busy in June and July, during the Australian school holiday period. If you visit in May, a number of the waterfalls may still be running, which results in some stunning scenery. The weather is cool enough for a light jacket in the evening but Kakadu generally experiences warm temperatures.
The wet season in Kakadu is from October to April. During this time, it can be very hot and humid. Some of the roads may be closed due to flooding, and there's plenty of insects around. If you pack carefully, you'll have a better experience of Kakadu in the wet season. The waterfalls are in full flood, you'll see wildlife everywhere, and accommodation rates tend to be lower. Pack long-sleeve shirts and trousers for the evenings as extra protection against mosquitoes, and don't forget mosquito repellent.