From the mining camp, head south on a narrow road for a few hundred feet passing a road or two that head to the north. Turn north on the last of these and continue until a use trail, marked with a cairn is encountered on the left side of the road, heading generally uphill and to the north. Follow this sometimes faint trail as it wraps around the north side of the lower eastern summit of Black Mountain. Once it reaches the north ridge, it continues on the west side toward the south, over scree slopes and through a small grove of bristlecone pines. The trail seems to end once it is above the saddle leading to the western summit. Follow this saddle and ridge west and then southwest to the western summit.
There are no sources of water, so bring your own.
Climber.org has a trip report from 1999. Though the report doesn't state it, they actually ended up at the lower eastern summit.