How can I backup my contacts on iCloud?

If you don’t sync contacts with iCloud, then they will be included in the iCloud backup. If you sync contacts with iCloud, they are not stored in your backup.

By syncing contacts with iCloud, I mean if you have turned on the switch to use iCloud with your contacts in Settings>Your Name>iCloud>Apps using iCloud>Contacts>On. When you do that, you keep your contacts in sync across all devices. However, when you sync contacts with iCloud, if you delete them on the device, you delete them in iCloud as well.

If you were to buy a new device and restored the backup, the contacts would not be recovered from the backup, so you would have to turn that switch on again in the settings and your contacts would download from Apple’s servers.