Open the Contacts app or the Phone app and go to Contacts
Tap the contact you want to delete
Tap Edit
Scroll down and tap Delete Contact
Tap Delete Contact again to confirm
Repeat for each contact you want to remove
To delete contacts synced from iCloud:
Go to Settings
Tap your name
Tap iCloud
Turn off Contacts
Choose Delete from My iPhone if prompted
To delete contacts synced from Gmail or another account:
Go to Settings
Tap Apps
Tap Contacts
Tap Contacts Accounts
Select the account
Turn off Contacts or delete the account
To remove all contacts at once from iCloud:
Sign in to iCloud.com
Open Contacts
Select all contacts
Delete them
To erase all data including contacts:
Go to Settings
Tap General
Tap Transfer or Reset iPhone
Tap Erase All Content and Settings