May is when things are greenest, waterfalls are flowing at their best, but in many years large parts of Kakadu are still closed well into June and some places even late into July due to flooding, the most accessible spots like Ubirr, Cahills Crossing will be open. July is the peak month for tourists, things are open, the weather is mildest, but it is also when it is at its most crowded. Early August the great exodus south starts as the weather starts to warm up, by mid August the big crowds are gone, everything will definitely be open, by September it is much much quieter but things can be starting to look fairly dry and the weather is getting hot. Personally I love the early season best but you really have to have the time to sit around and wait for places to open, late season I also like, July is simply way too busy overall for me. I assume if you are going to come all the way from Perth that you will be spending time in other places in the northern NT too, don't miss Litchfield, and Darwin area really has a lot to offer.