Signing in with Game Center will save your Kingdom Rush 5: Alliance's progress automatically. This means you'll be able to pick up where you left off, even if you uninstall the game or switch phones, as long as you use the same account and connect to the cloud.
You can check that your cloud save is enabled by following these steps:
On the Options screen, find Cloud Save and check if the option is enabled (it should appear this way).
If this is not the case, and you see Cloud Save with a red cross, please tap on it in order to enable it.