Moving contacts between groups cannot be done via iPhone. You will have to sync your iPhone via iTunes and ensure that contact sync in enabled. The following process can also be used with your GMail account instead of Exchange.
To move the existing contacts, follow these steps:
Connect your iPhone to your Mac.
Open iTunes and look for your iPhone under devices in the left navigation section.
Click on your iPhone to view iPhone sync settings.
Click on the Info tab at the top of the sync section.
On the Info tab, select the checkbox next to 'Sync address book contacts'
Click Sync on the bottom right corner of the sync section.
The above instructions will sync your iPhone contacts to your Address Book on Mac.
Open Address Book.
Add your exchange account, if you have not done it already.
After that you can move your iPhone contacts to your exchange account by dragging-and-dropping.
You should also configure the iPhone settings so that newly created contacts are created in the correct place, so you don't need to repeat this process,
Go to Settings -> Mail, Contacts, Calendars
Scroll to 'Contacts' section
Tap on the 'Default Account' tab and select your exchange account.
You should be set now!