If you have an old contact on your phonebook that you no longer need, you can delete it from your iPhone 4 smartphone. Here's how:
1. Go to the Contacts app.
2. Then select a contact.
3. Next tap on Edit.
4. Scroll down the screen and tap Delete Contact.
More Information on contact management from Page 211 of iPhone user manual.
If you cannot see the edit button then see the comment below from Julie:
"If any one is struggling to delete contacts because there is no edit button on the contact since the ios5 update on the iphone4 then u need to merge all contacts onto iCloud which can be found in settings this way all can't acts can b deleted like before the update. Hope this helps"
For merging contacts read iPhone 4: Merge Contacts to iCloud.