Backing up to iCloud and syncing data with iCloud are two different things. In order to sync your data with iCloud, go to Settings>iCloud and turn on the data you want to sync (such as contacts). If you've done this, but your contacts still don't appear on icloud.com, 0pen the contacts app on your phone and tap Groups. Do you have any non-iCloud accounts listed, such as On My iPhone, Gmail, Yahoo or an exchange account?
Also go to Settings>Mail,Contacts,Calendars>Default Account (in the Contacts section). Do you have the Default Account setting there, and if so, what is selected as your default account.