How do I set up an alias email with Mac Mail, so that my alias shows up when I send email from my Mac Mail account?

There are a couple of ways of doing this. The easiest is probably to just add the secondary address in the Mail.app -> Preferences -> Accounts field.

The specifics of which vary a little based on the OS/Mail.app version you're running, but in Yosemite:

Mail -> Preferences -> Accounts

Select your iCloud account and change the 'Alias' popup menu from your iCloud address to 'custom'.

Doing this enables the 'Email Address' field. In this field you can enter the email address you want to appear in your From header.

If you want to use multiple addresses (e.g. your iCloud address and your other alias), you can enter a comma-separated list of addresses - e.g. 'you@mac.com, anotheraddress@otherdomain.com' and then you'll have a pop-up menu on each new message where you can choose the address to use for each outgoing message. You can also set the default address in Preferences -> Composing -> Send new messages from:

Hi, I have exactly the same problem. However if I sign in to iCloud Mail. Click: Settings/Preferences/Accounts, I see my iCloud account, which offers me only my iCloud Email addresses. I have only 1 alias. I can change only the Label and the Full name. If I click on Add an alias it allows me to add a name but forces the iCloud email address to Name@iCloud.com. I have my own domain name and want the 'From' to say: my name <mail@mydominname.fr>

Can I do this?

Actually you can add email alias easily.

1. right **** on your email account that you want to add the alias
2. Click "Edit .....your email account"
3. At the Account Informations tab you can see the option Alias:
4. Click the dropdown menu and choose Edit Aliases then you will get to Edit and Add or Remove email alias.
5. Done

Next time when you reply email you will have option to choose to send as your alias email.