Pretty easy. Take Highway 168 east from just north of Big Pine. Head up toward Westgard Pass, coming to a steep grade and narrow road. No big RVs.
Come to the White Mountain Peak road, turn left (north) and follow signs to the Bristlecones. Essentially stay on the main road all the way, past where it turns to dirt at about Schulmann Grove. You'll come to the gate at about 11,000 feet. A good passenger vehicle can do this. Be careful to drive around any big rocks. and go slowly, as to not to bash your undercarriage on dips and rises in the road.
The only big junction of note heads right to the Naval Station. Go left or straight here. I can't remember if it was signed. I recall the total one way distance to the gate is some 22 miles (from 168). Don't hold me to this!
Good luck!
