If you use an email account to store your contacts, it will work this way:
Set up the account with your contacts on your iPhone. This has to be done in Settings - Mail, Contacts, Calendars. There, select "Add Account". If you use a Windows account (Hotmail, Outlook, Live), use Outlook.com or Exchange to set up the email address on your iPhone.
Enter your email address and password. When the account is set up, make sure "Contacts" is enabled.