There is little to do in Darwin. Maybe a day worth of stuff - but certainly the best stuff is out of Darwin.

I personally think Litchfield is overrated.

I would hire a car and head to Kakadu (about 3.5 hours drive from Darwin). My tip would be to do the Yellow Waters Billabong (in Cooinda) cruise at dawn. You can stay at the hotel where the boats leave from (otherwise it's a good hour drive from the Holiday Inn in Jabiru). Also the jumping croc cruises on the way into Kakadu are a bit kitch but really good. I've done both a number of times (with all my visitors) and it's really quite good. There is plenty to do in and around Kakadu National Park (particularly in July when all (or at least most) of the roads are open (closed due flooding most of the year). I would be tempted to do two nights if you can. Maybe one in Cooinda and another at the Holiday Inn at Jabiru.

July is the peak season - so expect lots of tourists. If you are coming in late July you will be running into Darwin Show, Darwin Cup and Darwin festival season and hotel rooms will be at an absolute premium (book early). The weather is absolutely perfect - so it evens out.

Katherine can be done after Kakadu (it's kind of sort of in the same direction). The gorge is nice - the town is a complete hole. No more than 1 night (you kind of have to spend a night there because it is a good solid 3 hours drive from Darwin). Accommodation isn't flash (especially in July when every man and his dog is looking for whatever half decent accommodation is going).