Note: GitHub Packages is currently in beta for GitHub AE.
You can only delete a specified version of a package on GitHub AE or with the GraphQL API. To remove an entire package from appearing on GitHub AE, you must delete every version of the package first.
Deleting a version of a package on GitHub AE
To delete a package version, you must have admin permissions in the repository.
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On GitHub AE, navigate to the main page of the repository.
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To the right of the list of files, click Packages.

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Click the name of the package that you want to delete.

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On the right, use the Edit package drop-down and select "Manage versions".

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To the right of the version you want to delete, click Delete.

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To confirm deletion, type the package name and click I understand the consequences, delete this version.

Deleting a version of a package with GraphQL
Use the deletePackageVersion mutation in the GraphQL API. You must use a token with the read:packages, delete:packages, and repo scopes. For more information about tokens, see "About GitHub Packages."
Here is an example cURL command to delete a package version with the package version ID of MDIyOlJlZ2lzdHJ5UGFja2FnZVZlcnNpb243MTExNg, using a personal access token.
curl -X POST \
-H "Accept: application/vnd.github.package-deletes-preview+json" \
-H "Authorization: bearer TOKEN" \
-d '{"query":"mutation { deletePackageVersion(input:{packageVersionId:\"MDIyOlJlZ2lzdHJ5UGFja2FnZVZlcnNpb243MTExNg==\"}) { success }}"}' \
HOSTNAME/graphql
To find all of the packages you have published to GitHub Packages, along with the version IDs for the packages, you can use the packages connection through the repository object. You will need a token with the read:packages and repo scopes. For more information, see the packages connection or the PackageOwner interface.
For more information about the deletePackageVersion mutation, see "deletePackageVersion."
You cannot delete an entire package, but if you delete every version of a package, the package will no longer show on GitHub AE.