If you change your default to icloud under settings>contacts the next contacts you add should go to the server and you can check by going to www.icloud.com.
If they are currently on your exchange account server and you want to have them on the icloud server you'll need to find a way to export them from exchange so you can then import them to icloud.
You can either go to your email web page server and export them from there or use an app from the app store to export contacts (my contacts backup app works well but backs up all contacts from all accounts).
On icloud.com you can import files in vcard format (.vcf) only. For what i know exchange accounts can only export in .csv format but for instance gmail can import .csv and export as .vcf.
More info here:
iCloud: Import a vCard
Get help using iCloud Contacts, Calendars, or Reminders - Apple Support