With a good 4WD car (Jeep, FJ, Xterra, etc.) a competent driver can make it almost all the way to the start of the Castle Peak trail. Most people will have to park much further down, however, especially during the spring and early summer. During the winter, the road is closed just beyond the lowest point mapped.
This trail is too steep and rocky to truly run. There are plenty of much better options in the area.
From the upper trailhead at 12,800 ft (see Castle and Conundrum
; top mapped parking), run into wide open Montezuma Basin. From here, follow a poorly marked and difficult, rocky trail to the left. Your goal is to run up to Castle Peak's northeast ridge. Don't fret too much if you can't find the trail. Aim for the saddle in the ridge and you'll be fine. Run up steep talus to 13,400 ft, at which point the ridge is mellow.
Continue up 300 ft further along the right side of the ridge before reaching the ridge proper (13,700 ft). Things become more difficult from here. From here, the grade is very steep and the rock very loose. It doesn't quite require scrambling, but it's close. Fortunately the trail here is a little clearer, at least for a little bit, so you can't get too off route.
Run/scramble up the remaining 500 vertical ft of loose, steep rock. The summit is visible from about 14,000 ft, so don't get too amped up for the false one. After the false summit, drop just a bit to a small saddle. The summit's less than 300 ft up. Push on.
From here, turn back, or continue on to Conundrum Peak
Up this high, there's not much other than marmots and pika.