Follow the Auriol Trail to the first main junction (there's a 2 km trail marker there). Continue on the right trail, which is slightly shorter and more direct for the Martha Black route. The Auriol Trail is well maintained and is within the forest for the most part, until you get closer to the upper part of the loop near the backcountry campground where you finally reach the subalpine and get some views of the Auriol Range.
About 500 m before reaching the Auriol Backcountry Campground, you will leave the main trail and follow a creek up into the alpine - towards the moraines. Pick your way along the creek, crossing as necessary to avoid any bushwacking. As you approach the base of the main rocky moraine, you'll need to follow along the right edge of it, along a small narrow drainage. Depending on the time of year, there may or may not be water flowing down this drainage and there may be some snow patches remaining into the summer. The drainage opens up as you gain more elevation. After about 1.6 km after leaving the Auriol Trail, you'll need to get up on the slopes on the right. You should be above most of the willows and shrubs on the slope and on a nice alpine plateau. The plateau offers many great areas to set up a camp - there should be water available in the creeks in the surrounding drainages and potentially small melt water channels off any snow patches.
Traverse the plateau contouring along the base of a large pyramidal mountain (on your left). After about 1.6 km of heading northwest along the plateau, you'll drop down to another small creek drainage, and then back up the other side to continue on the next section of alpine plateau heading east. The wildflowers eventually give way to rock and scree as you get closer to the next glacial valley. Continue contouring around the base of the mountain, following up the glacial valley (now heading southwest). Do not drop down too low or too close to the creek, making use of the various ramps and ridges along the moraines. After just over 1 km of following up the glacial valley, you will need to get to the other side of the creek. This is just before the creek upstream heads underground for a short distance.
Continue heading south and now up the moraines towards the base of the large mountain ridge in front of you. Note glacial ice underlies the rocky surface of the moraines, including some obvious crevasses in some areas, so keep clear of the steeper, more gullied morainal area to the right, staying more to the middle, on the more gradual ridge part of the moraine. The terrain eventually levels out slightly as you continue following the glacial valley gradually uphill to the right (west). There should be a few small, meltwater creeks in this area where you can refill on water.
You will eventually reach the base of a steeper snow and ice section. Take a gradual route up staying to the right of the steeper exposed ice patch. Crampons may be helpful if the surface is firm. It is advisable to rope up for glacier travel for this section as this small glacier is crevassed. It's just about 100 m elevation gain till the glacier levels out again. Keep roped up and continue up the glacier towards the base of a rocky saddle. Stay clear of the mountain sides as rock fall is common in the area. Once on the rock again, you can remove the rope and climb up the scree slope (140 m) to the top of the saddle. Once at the saddle, head west (left) on the ridge to the first steep rocky section. At this point there's a 10-20 m section of of Class 3 to 5
climbing/scrambling. This is the most exposed and steepest part of the climb (a rope may be comforting for some), with lots of rock fall hazard (helmets are a must). Extreme caution is necessary.
Once up this crux, it's a short hike up a steep scree and snow slope to a wide open rocky plateau with the final view to the summit. Traverse the plateau till you reach the base of where the glacier and snow start again. Switch over to 'glacier travel' rope-mode again for the final approach to the summit. Take the more gradual route along the left ridge to the first sub-peak and then up to the summit. Enjoy the views of the surrounding peaks of the Auriol Range, and beyond to the Alsek Valley and the St. Elias Mountains.
Return the way you came. It's a little quicker on the way down than the way up, as with most mountain hikes.