Mid-June, even assuming snow continues to stay at normal levels rather than way above normal like last year, is pretty early. You are going to be doing a lot of walking on snow. Check the snotel sites (remember that those are inches of water equivalent, not inches of snow) starting in mid May. Page down to the Upper Green River Basin section and look for Elkhart Park.
Streams will also be high with snow melt, which will be a problem some places. Note that in the Winds, the USFS builds bridges only where it is unsafe for horses to ford in low water! You don't even want to think about trying to cross Pole Creek in high water. You can check stream flows at:
Click on "Pine Creek above Fremont Lake, WY"
You really should to take a couple of days to acclimatize before starting a trip at 9400 feet that goes up to 10,500 the first day, which is what happens at Elkhart Park. Everyone reacts to altitude differently--you may have no trouble at all going right up to 11,000 feet, or you may start getting symptoms at 8,000 feet. At least stay one night in Pinedale and then go up to Elkhart Park, dayhike around and then camp there. That will give you a pretty good idea whether you'll have problems the first day on the trail. Plan short days for the first 2 or 3. Most important, read up on the symptoms of Acute Mountain Sickness so you recognize them if they happen!
Frankly, for mid-June I'd suggest Yellowstone instead. The altitude is lower, so you can start right in if you take easy days. For the same reason, most of the snow will have melted. The Park staff will be happy to work with you to plan a good trip and give you the appropriate permits. Save the Winds for a future year when you can come in July, August or September. There are low altitude hikes in the Winds, but you'll be in a lot of bark-beetle-killed timber. The Winds are at their best near and above timberline, which is most of the wilderness area.