June is a the best month for boating and the entire state will be running. Judging from the snow we have already got, its going to be a great season!
With a month in the state you will have enough time to travel to all over the state.
I would start looking through Kayaking Information
to check out runs, descriptions and difficulty. This can give you an idea of what you want to hit.
The new guidebook, whitewater of the southern rockies, has loads on info in it and would also help you plan your trip.
There is a lot of class IV right around denver on clear creek, boulder creek, the north fork of the south platte (bailey). There are also lots of boaters to hook up with around denver. It would be a good start to the trip.
The arkansas (salida or buena vista are the towns) is also a great stop for boating. Big congregation of boaters, lots of different runs to do, and easy to hook up with partners. Camping out is the norm, but it gets cold and night, so bring warm gear too.
The roaring fork valley around aspen has a class IV gem, the slaughterhouse run.
Durango class IV upper animas.
If you will rent a vehicle and your own gear, get the guidebook and plan on cruising around the state to get the best water.
If you wanted to stay in only one place, the arkansas is a good bet.