From Conrad Kain Hut, ascend the moraine to the right side of the Crescent Glacier, then walk the glacier (It's almost flat.) to its head under the Bugaboo-Crescent saddle. This part of the glacier is so flat, it's almost devoid of crevasses. Almost. I haven't seen climbers rope up for glacier travel on this approach, but accidents have happened.
Climb class 3 to the saddle, turn left and walk and climb easy rock to the large, rubbly ledge that crosses the ridge. Enjoy the views of the Vowell Group to the north while you dig the rock gear out of your pack.
Begin on steep, clean flakes just left of the crest, and climb to the top of the flake/pedestal to belay. Descend left to flakes that take you up and out onto the E. face for the second pitch. From the top of the flake system, catch a fat, white quartz sill that takes you up and right around the ridge crest, then climb up to belay at a roomy ledge. The fourth pitch starts up a clean finger crack that 50 Classic Climbs calls the crux. (I thought the quartz sill just passed was trickier.) Pitches 5 & 6 are actually in a blocky gully right in the crest. Straightforward and enjoyable. As the ridge levels off it's still sharp, so there's no routefinding involved until near the summit. The distracting views are the main difficult. Green and Benson say you've climbed 10 pitches when you reach the boulder that blocks the ridge. Find the anchor at your feet and rappel the left (E) side about 20 feet (6m) to a rubbly ledge. Carefully follow the ledge to just below the summit and tunnel up between blocks.
Descnd by downclimbing to the slabs that take you to the South summit, then descend the Kain route.
Carry a full rack for sustained free technical climbing on clean, solid rock. Include longer slings to reduce rope drag. In June you can probably get to the start of the route without an ice axe, but you'll need it descending from the Bugaboo - Snowpatch col. Later in Summer you'll need crampons. You've probably figured out by now that you're not caching any gear the the start of the route. If you climb well in lightweight trail boots you may be able to wear them the whole way and not have to carry boots in your pack. The Kain route has some solid rock, but much of it will be hiking down steep rubbly slopes. Wear a helmet. Bring clothes for storms. I've been hit by snow below 9,500' (2,900m) in high summer.
When descending the Crescent Glacier after your climb, stay away from Snowpatch unless you're roped and prepared for crevasse rescue.