North Dome is a granite dome directly opposite Half Dome 3,500 feet above the eastern end of Yosemite Valley, and is one of the most spectacular places I've ever hiked. Your payoff is looking Half Dome eye-to-eye, and it is indeed worth the effort. In fact, most of the physical exertion on this hike is on the way back to the starting point at Porcupine Creek Trailhead.
This entire hike has metal signs at every junction so route finding is exceptionally easy.
The trailhead is on the south side of the Tioga Road (Hwy 120), and begin by heading downhill on an old road for a few hundred yards until it turns into the signed Porcupine Creek Trail at a junction with trails heading due east and west respectively. Continue straight/south through a forest of majestic red fir. These are almost identical to Douglas fir, but the bark is reddish. Think of a ponderosa pine's color, but more brown than red. The hike continues mainly flat to the south for a mile-and-a-half before reaching another junction.
Take the middle trail heading due south 1.3 miles through more red fir with increasing numbers of sugar pine and Engelmann spruce as the trail follows the contour on the east side of the Lehamite Creek drainage to the junction with the trail that heads up to Indian Head Rock. The top of Indian Head has an arch that is supposedly pretty cool, and the trail is a little more than a quarter mile each way. This section has a nice uphill stretch through a lot of downed trees before reaching the junction. Continue south and downhill into the upper reaches of the Royal Arch Creek drainage before the trail heads uphill for a bit through a forest that grows increasingly drier and more open. Widely-scattered Ponderosa pines dominate the top of the ridgeline that the trail has reached. The trail continues south and slightly downhill through this open landscape with intermittent views of Half Dome and Clouds Rest, but don't take too much time here for pictures because the real show is further ahead.
The trail reaches a junction about a mile from the Indian Rock junction, and the trail heads left very steeply downhill before switchbacking to the right/south directly to North Dome. This half-mile section has some spots that might require some hikers to do some butt-scooting, but it's not too difficult. The trail continues along level terrain after the switchback through a small but deep mixed conifer forest before everything begins to open up. Continue as the trees become stunted and widely scattered Ponderosas uphill to the top of North Dome.
The incredible view of Half Dome from the summit is immediately apparent upon reaching the top. There is a nice cluster of weathered rocks that form a great break spot at the highest spot, but one can travel a few hundred yards south closer to the edge on the south side for more views. The view of Half Dome is impossibly beautiful, but Clouds Rest to the northeast isn't far behind. I prefer the view of these two massive hunks of rock to the east so I'd recommend doing this hike in the afternoon when the sun is at your back. Views to the west of Yosemite Valley, El Capitan, and the foothills beyond are still impressive. The high peaks of the Cathedral and Clark ranges are visible high to the east and southeast respectively, and typically hold snow well into summer if not all year. Needless to say this is a photographer's dream.
Enjoy the top of North Dome then return the way you came. There is more uphill than downhill on the return trip with most of the tougher portions closer to North Dome.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.