IPhone – Just turned on icloud for phone, but moved contacts from Outlook on local machine

Yes, it is possible.

Set the default account for Contacts to iCloud. That will add new contacts to iCloud.

To do this

1. Tap Settings -> Mail, Contacts, Calendars
2. Scroll to Contacts, about 2/3rds down the page.
3. Tap Default Account and then choose iCloud.

You might be facing the synchronization problem:

Microsoft chose not to support CalDav and CardDav in Outlook for Mac, unfortunately these are the services required by iCloud.

Although MS did add these to Outlook for Windows they did not add them to Outlook for Mac.

In short you can't subscribe (not sync) to iCloud from Outlook for Mac (although you can if you use Outlook for Windows)

Use Outlook, or use iCloud, not both.