Other than to explore the abandoned old downtown and a few scattered artifacts, there isn't much to do in Clifton except take photographs and say you've been there. But there is a reason to go to Clifton.
Clifton is the northern entrance to the Black Hills Back Country Byway, a delightful couple-hours drive through the rugged beauty of Greenlee County that takes you from Clifton to Soloman on Hwy 70 near Safford. In fact, before modern highways, this unpaved 21 mile scenic drive was the only road from Safford to the mines at Clifton.
The Byway crosses the Gila River several times and offers travelers many campsites and picnic areas. The campsites are barren places with a ramada, BBQ, and firepit.
The picnic areas are completely different. Some offer magnificent vistas. Others are in thick riparian areas along the Gila, particularly at Gila Box. You can see for yourself on this brief slideshow.