When I open Contacts app in iPhone 5c, I see one contact displayed on the top of the list with bigger font and icon.
How to choose this entry?
The contact displayed on top of the list with bigger font and icon is generally your own contact entry, i.e. the contact entry corresponding to logged in Apple ID.
You can tap to select and navigate to the details screen and edit the fields in the detail screen.
Update: As mentioned by @fsb in comment, Go to Settings app → Contacts and verify the contact selected for My Info is actually your own contact card. That's where Contacts app gets the name for the top of the list entry.