How do I install iCloud app on my iPhone

What exactly do you mean by "iCloud App".

iCloud access is built in on iPhones. All you need to do is log in to iCloud, in Settings->Account(top of settings), and make sure you are signed in to iCloud there.

Then you can use all the Apps that access iCloud, such as Contacts, Notes, Photos (if you Photo Library turned on), Calendar, Pages, Numbers etc...

If you want to access iCloud Drive for files you can use the Files App.

Use the Files app on your iPhone, iPad, and iPod touch - Apple Support

There is no app with the name "iCloud." There used to be an "iCloud Drive" app but that is now the "Files" app. If that doesn't help, you may want to reply with more details about the app that you are asking about.