From Egilsstaðir town, the major town of East Iceland located by Highway 1, there are two routes to Hengifoss. You can drive on either the west side or the east side of lake Lagarfljot. The distance is the same, around 35 km.
If you pick the west side you can expect gravel road for part of route. You will find the junction with Highway 1 on the hill by the timber bridge close to the Egilsstaðir airport. It is marked as Route 931 with signs pointing to Fljótsdalur and Skriðuklaustur.
The route on the east side of the lake is more popular passing the beautiful Hallormsstaður forest and with great panoramic view to Mt. Snæfell, which is inside the Vatnajökull National Park. From the junction of Highway 1 and Route 92 in the town of Egilsstaðir, follow Highway 1 (the Ring Road) south for 11km, then continue straight along the road as it becomes Route 931 where Highway 1 bears to the left. Continue along Route 931 for another 23km, to a T-junction after a long bridge from where you can admire the waterfall falling from the plateau. At the T-junction bear left where signs point to Snæfell and Skriðuklaustur. The parking lot for the waterfall is just a little bit further, by the bridge crossing the Hengifossá.
From the parking lot there are only 5 km to the visitor center of Vatnajökull National Park at Skriðuklaustur where you can also find a culture centre, heritage site and great restaurant. Just continue on Route 933 further up the valley.