The Alpine Loop is normally open by June, but this year there are concerns. The following website says they are hoping for the middle of June. I encourage you to bookmark and check often:
Ouray Jeep Trail Conditions.
In a nutshell, June is a bit early to have the run of the place. Many things will not be clear of snow and avalanche damage. Note also the following numbers for trail conditions from Silverton's point of view:
Silverton Visitor Center: 970-387-5654
Silverton Public Lands Ofc? : 970-387-5530
The Alpine Loop consists of Engineer and Cinnamon Pass trails. At the link above you'll want to look for those names. Engineer is the better of the two, but Cinnamon is likely to be open sooner because locals use it. While Cinnamon is certainly better than driving on pavement, in my opinion it's no where near as amazing as many of the others in the area.
I notice you have 5 days for your trip. Frankly, I would want 5 days just for this Colorado area. If you were going to be there late enough that everything was open I'd strongly suggest you do Imogene Pass and others plus Black Bear if you're really adventurous.
There are others here that can probably give more in depth advice, but don't hesitate to ask if you think I can be of any service. BTW, I've been interested in a Utah Traverse. Do you have a link for info on the route you are taking?