1. Make sure your Apple ID is correct when you go to Settings > iCloud on your iPhone 5S. If it is not sign in to your Apple ID.
2. Make sure contacts is toggled ON.
3. Make sure contacts is toggled OFF in other servers you might use for email like Outlook, Yahoo, Google and similar. Have a look account by account in Settings > Mail, Contacts, Calendar > each individual mail account. Syncing with more than one server some times creates conflicts.
That should be enough to sync your iCloud contacts to the iPhone, provided your iPhone is connected to Wifi or cellular data.