The road actually does not go completely to the top but rather, stops a hundred or so yards short. An unmaintained trail goes up the remaining way.
For most experienced climbers, the view isn't exactly what would be deemed "impressive" although the city of Palm Springs is visible to the southwest. Due its modest prominence of 776 feet, the view is obscured by other mountains to the east and north.
Take CA-62 north from the junction with I-10. In Yucca Valley, turn right onto Sage Ave. and immediately left on Yucca Trail. Continue straight until buildings give way to open desert and watch for a dirt road on the right. There should be a paved road opposite it. Entrance is unsigned as of early 2009 (unless I just missed it somehow). Turn right and head straight through what looks like an isolated community.
Continue through (on La Contenta Road) until the road forks left with a sign for Covington Flats. Take the fork and continue past the park entrance sign through a grove of Joshua Trees. A second fork should come up with the left fork having a locked gate entering the flat. Continue with the right fork. Continue to a sign for Upper Covington Flat and turn right. Head straight on this road past the turn for Upper Covington Flat (another one) at which point the road slowly begins to ascend the peak. Park at the end of the road.
RegulationsA $10 7-day park pass is required and may be paid at the Black Road Canyon visitor center. Campgrounds are found there as well. Annual pass is $40 good for Joshua Tree NP only or $80 for all federal lands.
Alternatively...The peak may also be climbed via a 10 mile route from Black Rock Canyon on the California Riding and Hiking Trail and the Eureka Peak Trail. The route ends at the parking lot.
External LinksJoshua Tree National Park on NPS - official site of Joshua Tree NP.
Biking Eureka Peak - more information regarding biking up the peak.
Eureka Peak on Wikipedia - some general information on the wiki.