Go south from Tower on the narrow blacktop past Tower Falls to the start of the Chittenden Road (the trailhead) and overlook. Alternately, the same point is reached by driving north from Canyon (towards Tower) and over Dunraven Pass. The Chittenden Road to the summit was once (amazingly) open to the driving public. Get an early start. If someone decides to stop in the middle of the highway (a common, frequent occurence in YNP) the resulting traffic jam can delay you by as much as two hours.
This road is used to access the 2-story lookout building on the summit. The public can hike it or mountain-bike it. Follow the road as it switchbacks for 2.5 miles passing the junction with the trail from Dunraven Pass (a slightly shorter route from the currently closed pass) and on to the summit. This hike to the highpoint of the Washburn Range is a very popular ascent to the most central summit in the park with views of the following ranges:
Teton, Centennial, Henrys Lake, Madison, Gallatin, Absaroka, Beartooth. Visible are Granite (Montana's highest) and Grand Teton (second highest in Central Rockies) as well as the entirety of the Yellowstone Plateau and Washburn caldera.
Dayhike. This area is heavily frequented by grizzlies and scat/sightings are common. If you find yourself here with no other hikers (like, say, October under bad weather) prepare accordingly.