The other trail that you mention is probably the High Divide trail. It is a more direct approach to the Adams glacier and the start of the climb. You would bypass High Camp. Head up Adams Creek from the jct with the PCT.
If you take the Killen Creek to High Camp approach, you'll need to traverse west onto the lower Adams glacier and over to the right side to the start of the climb. Since most climbers descend the North Cleaver, setting up a base at High Camp or above is preferred.
Usually the 2329 road is not clear of snow until early July. In early season, its best to approach the Killen Cr TH from the Randle side (23 road) because the road holds late snow between Takhlakh Lake and High Divide TH. The Forest Service office in Randle will have the best information on road conditions.