Burro Pass can be reached most easily from Twin Lakes via the Barney Lake trail. Hike along the Barney Lake trail past its namesake lake to the intersection with the Rock Island Pass trail, and follow the latter trail past Crown Lake south to the Burro Pass trail. The Burro Pass trail crosses over Mule Pass (10,440 ft+) and passes through the upper reaches of Slide Canyon to reach Burro Pass (10,640 ft+).
Burro Pass can also be reached from the south via Matterhorn Canyon. This option would probably be feasible only as part of a longer backpacking trip through northern Yosemite.
From Burro Pass, head east around/over a small rock outcropping (shown on the USGS 7.5' topo) and ascend granite slabs towards the broad gully leading down from the low point along the ridge just south of the summit. Occasional ducks show the approximate route near these slabs, but the route is easy enough to find without religiously following them.
The gully itself is mostly a mixture of sand/scree and no more than class 1 until the ridge top is reached. From the low point (which is also the likely target if ascending the peak via the SE slope), head north to the summit. For this final portion, it's easiest to stay directly on the ridge top.
Because of its sandy nature, which would probably make for a bit of a slog if climbing the peak, the SW slope is perhaps best used as a descent route. It can be combined with an ascent from Twin Lakes via the East Couloir or SE slope to make for an interesting loop trip around Mule Pass, doable in a day.
None for the route during summer months.