The best trailhead for the hike is the Highline Trailhead located along State Highway 150 33.2 miles east and north of Kamas, or you could drive south from Evanston. You could also start at Mirror Lake, but you must pay to park there, and there isn't as much room to park.
From the trailhead at 10,347 feet, follow the Highline Trail .8 miles to the junction with the Mirror Lake Trail. The trail continues south and east and drops to 9900 feet. Scudder Lake is passed after 2.2 miles (from the trailhead). After another .8 miles, pass a junction with the Wilder Lake Trail. Stay left on the Highline Trail. After another 1.3 miles, stay right at the junction where the Naturalist Basin Trail branches left. Continue on the Highline Trail towards Rocky Sea Pass, passing the junctions with the Pinto Lake Trail after 1.5 miles and the Four-Lakes Basin Trail after another 1.2 miles. Rocky Sea Pass is another 1.2 miles along a steady up hill grade. The pass is 8.2 miles from the trailhead. Enjoy the spectacular views.
Continue east and down the steep east side of the pass for 1.9 miles to a junction. From the junction, turn right (east) while staying on the Highline Trail.
(You can also head left (north)at the last junction and use several variations of the Head of Rock Creek Trail to reach Ledge Lake (see below). These routes are longer, but more scenic. Lightning Lake is my favorite area in the Rock Canyon Basin).
Continue on the Highline Trail east for 1.2 miles to a 4-way junction, just after the bridge over Rock Creek, keep straight on the Highline Trail. After another 2.2 miles, the trail reaches the east end of the Head Of Rock Creek Trail. Stay right (east) along the Highline Trail and follow the trail for 1.5 miles to Ledge Lake.
From Ledge Lake, the west face of Explorer Peak is highly visible, and you can visually pick out a route up the mountain. There is a junction at Ledge Lake. Leave the Highline Trail and turn right (south) on the Squaw Basin Trail towards Phinney and Anderson Lakes. Follow the trail for about .8 miles until you are west of the peak. From there, head east to the timberline and base of the peak (.9 miles). You can climb Explorer directly up the west face, but be prepared for some scrambling over some massive teetering boulders. The final part of the route climbs 1700 feet in only .5 miles and is very steep.
The round trip distance to the peak and back is about 35 miles and should take 3-4 days, unless you want to linger around the lakes and other peaks along the way.
Map, compass, and a good pair of boots is needed. This is a 3-4 day trip so backpacking gear is needed as well.