Starlight Peak is but a sub-peak of North Palisade, but due to it's spectacular summit block (the "Milk Bottle") and the difficulty of most routes to its summit, it has acquired an aura of its own, and justly ranks among the 15 California fourteeners. The most common way to climb Starlight is to traverse from North Palisade, or (less frequently) from Thunderbolt Peak.
Drive to Big Pine in Owens Valley and then drive to Glacier Lodge. Hike the North Fork of Big Pine Creek for approaches to North-East facing routes.
Alternatively, drive to Bishop and then to South Lake; hike over Bishop Pass and into Dusy Basin. Cross Thunderbolt Pass (cross-country) into the Palisade Basin. This is the approach for South-West facing routes.
Overnight wilderness permits are required at all times, and both both Bishop Pass and the North Fork of Big Pine Creek have use quotas in effect from May 1 to November 1.
Permits can be obtained from the ranger stations in Lee Vining, Bishop or Lone Pine. If under quotas, make reservations is advance. More information can be found at the Inyo National Forest Visitor Center.
Check mountain-forecast.com for current conditions, or call the Inyo National Forest Rangers at (760) 873-2400 for conditions. Daily report are available at the ranger station.
Another nice source for weather data is Howard Sheckter's webpage out of Mammoth Lakes.
See Stephen F. Porcella & Cameron M. Burns, Climbing California's Fourteeners.