If you want to do the entire clove it is easiest to hike up the stream, and then back down the road. Many people prefer to do the clove in two halves as there is a large waterfall in the middle that is difficult to negotiate around. Spotting a second car (or bike) at the top is another option.
It is easy to visit two waterfalls in the Platte Clove Preserve near the top of the clove. At the Long Path trailhead leading south from the Platte Clove Preserve there is room for about 4-5 vehicles along the Platte Clove Road. There is also an off road parking lot 0.2 miles east at the trailhead for the Long Path going north toward Kaaterskill High Peak. On the Long Path going south you will find about a small 5' falls a short distance east of the kingpost bridge carrying the Long Path over the Plattekill (use the south side of the stream). A more impressive 50' waterfall is reached on a separate 0.2 mile trail which starts just east of the Long Path trailhead, down a driveway to the Preserve's artist-in-residence cabin. From that waterfall you can venture down the south side of the Plattekill toward more waterfalls. It is very steep and dangerous. As I recall there are three more waterfalls, at the third you will reach the Hell Hole Brook which enters from the north.
An alternative way to this point is to descend down the west side of the Hell Hole Brook from the stone bridge on the Platte Clove Road. This is just east of the Long Path trailhead going north (site of the off road parking mentioned above). On the Plattekill below the Hell Hole Brook confluence you will shortly arrive at the top of the highest waterfall in the clove. From this point is nearly impossible to descend.
The lower clove is reached by following the Plattekill from the base in West Saugerties. For
road directions from exit 20 (Saugerties) of the Thruway refer to this webpage. Not mentioned in that description is that Route 33 turns left/south toward Woodstock at a sharp bend. Don't take that left turn, go straight ahead toward the mountains. At the point where "the road – now known as Platte Clove Mountain Road – begins a steep two-mile ascent through Platte Clove", you will find a road on your left (Tissell Road, I don't think there is a road sign though). Turn left here, on the right along a tall fence there is room for about 4-5 vehicles to park before the bridge over the Plattekill.
The land on both sides of the creek at Tissell Road is private and posted "No trespassing". Walk up the Platte Clove Mountain Road until you reach state land (about 0.4 mile). There is not much in the way of parking on this steep road - so it is easier to park on Tissell Road. From the road it is a very steep bushwhack descent to the unmarked path. The path will lead you past about 4 waterfalls. After the path ceases you may occasionally find better footing on the south side. Many places it easiest to carry your footwear and walk in the stream. When you reach the largest waterfall (mentioned above) the best way to get the upper clove is to scramble up the steep south side of the clove. In doing so you will climb well above the top of that largest waterfall before you can safely contour back to the west to rejoin the stream. You may end up a couple of waterfalls above the confluence of the Hell Hole Brook. If you want to take in those falls you will need to descend down the stream, and then back up to the top of the clove.
To re-emphasize the hike is very steep and dangerous in several places. Also the property owners in West Saugerties have long been upset with trespassers
(refer to the customer comments for a book which dared to even briefly describe the lower clove). Another book, Peter Kick's Catskill Mountain Guide (AMC) describes the 0.2 mile trail to the 50' waterfall at the top of the clove.