To remove the iCloud account with your old email address do the following:
1. Go to Manage your Apple ID and sign in with your current Apple ID and password. EDIT the Apple ID back to the old email address. This is temporary. Do NOT verify it, do NOT change the password: Apple - My Apple ID
Note: This will only work if the old email addess was used for this same Apple account (in other words, you updated that old ID to the current one rather than opening a brand new Apple account. If you opened a brand new Apple account, then go to the instructions below these.
2. Go to Settings>iCloud and scroll to the bottom. Sign out the iCloud account and type in your current Apple ID password when prompted
3. Go back to Manage your Apple ID and sign in with the old Apple ID and the current password. EDIT the Apple ID back to the new email address. You may need to verify it again
4. Go back to Settings>iCloud and sign in with the new Apple ID
If the old Apple ID is actually a separate Apple account, then you can still sign into Manage your Apple ID with that email address, and click on Reset Password. If you don't remember the answers to your Security Questions, you can contact Apple Support to have them reset:
ACCOUNT SECURITY CONTACT NUMBERS
Cheers,
GB