How do I go about requesting all my icloud data be deleted and cleared permanently?

Nobody else has access to your iCloud so yes - it's up to you.

However, it isn't "iCloud data" because iCloud is a syncing service - iCloud unifies the data across all your devices. If you remove data from iCloud it will also be removed from your devices. If that's what you want go ahead, but if you want to keep your iCloud Drive files on your devices, for example, leave them where there are.

In Manage your iCloud storage Apple offers some suggestions about managing your use of iCloud space. You might find some tips there.

In the case that your intention is to be “forgotten” by Apple, please consult with the following documentation.

Understand and control the personal information that you store with Apple

1. Get a copy of the data associated with your Apple ID account - Apple Support
2. How to deactivate your Apple ID account - Apple Support
3. How to delete your Apple ID account - Apple Support

"How do I go about requesting that my icloud saved/copied data be deleted and cleared? Can I do this myself? on both my iphone and Laptop."

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Clearing iCloud:

Yes. Do so on a laptop. Download all of iCloud, and then empty iCloud. See my user to on how to download it all: Getting a Quick and Direct "iCloud Drive-to-External Hard Drive" Download - User Tip

You omitted the word tip, as in user tip, in your response.