Park at Stanislaus Meadow (or Mosquito Lakes, but the fomer is a nicer hike) and hike out on trail 19E02, crossing a sometimes-dry creek (don't lose the trail here!) at 1.2 miles. Continue, climbing steadily for the next 1.1mi until you reach a junction with the Mosquito Lake trail 19E94. Keep right, staying on 19E02 for another 1.3 gently ascending miles until you reach Bull Run Lake.
From Bull Run Lake (an excellent camping spot), head SSE and hopefully pick up a use trail heading up a steep meadow to a ridge. Upon attaining the ridge, you will see some flat-topped rock formations to your left. This is not Bull Run Peak (I believe its Peak 9352). Contour in front and to the left of this formation, then follow the obvious ridge north until you see Bull Run Peak to the northeast (its pretty obvious!). Don't make the mistake we made and try to make a bee-line for the southeast flank unless you want a prolonged boulder-hop with ravines a-plenty. Instead, just follow the ridge to Bull Run's west flank and skirt its talus pile, heading east.
From the southeast side of the mountain, head more or less straight up the talus pile, keeping to the left (west) of some sheer rocks on the east end of the summit plateau. See the picture for details.
Once on top of the plateau the true summit is an easy walk to the west. When I last visited there was no summit register, but rather a cairn with a steel plate on which was scratched a message indicating that the register has not been destroyed, but is in the safekeeping of somebody in the 209 area code. Weird.
No special gear is required other than the usual dayhiking stuff.
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