Depends on your month of visit. If you are planning a trip during summers then you can try camping near Chail. This town is around 130 km from Chandigarh and you can go to Kufri from Chail on next day, route to Kufri form Chail is very less travelled by tourists so you will love the views and then from Kufri you can to Shimla and from Shimla to Chandigarh back. In this way you will cover a full circle .
Biking from Manali to Leh. there are numerous camp sites, where you can set up camp and rest or even stay 1-2 days if the views of the place are promising enough. Take the route that takes you via Tso Moriri and Hanle. These two places are a must see if you are serious about biking to Leh
You can try Auli. it's picture perfect to enjoy by yourself Bike ride and self camping is a must. there are a lot of camping spots on the way too. do let me know if you want to have a look at some of my road trip pics to auli from 2 years back.