From Durango, CO, drive west on US160 to the small town of Mancos. Turn right at a main and marked intersection and go about .4 miles north, then bear right onto CR42, which later becomes FR561 as it enters San Juan National Forest land. Drive north on this road about 10 miles to an intersection for the "Transfer Campground." The first four miles or so are paved with the remaining 6 being graded gravel. Coordinates for the intersection are: N 37° 27' 58.77" W 108° 12' 38.59". A right turn will take you shortly to the campground, but to reach the Sharkstooth TH, continue straight for another 1.0 mile to yet another intersection where you will bear right. In another .9 mile, you should come to the turnoff for the "Aspen Guard Station." Continue straight east here for another .5 mile, still on FR 561 and turn right onto FR350. Follow this road about 7 miles generally east to another intersection for Twin lakes. Turn right here onto FR346. Drive another 1.6 miles to the trailhead parking area. This last 1.6 miles to the trailhead will become rougher and a good clearance vehicle is advisable, though 4WD is not necessarily required.
The nearest National Forest campground is the "Transfer CG" about half way in on the drive. Here is the link: https://www.fs.usda.gov/recarea/sanjuan/recarea/?recid=43262. The site has potable water and vault toilet. There are about 12 sites and also a group area that can be reserved.
The remainder of the drive in to the trailhead offers several primitive sites and it's possible to car-camp at the trailhead, but there's little privacy if others are present. There are at least two primitive sites near the Twin Lakes and another primitive site past the lakes on the way to the trailhead.