You can't login using an alias, because it's not an email account, just a pointer to a real account. If you want to login using it, you need to create an account with the same name (and had to delete the alias first).
But all if you need is to send an email with that address as from address, you may setup the email client software accordingly. It is possible to configure more than one optional email addresses, and one of them can be selected as sending address while sending email.
For example: here is How to send from other email address in Outlook 2013. But there are some issues that needs to be noted (quoted from above link, applicable in general):
Issues sending from other email address
Please note that:
such emails sent from other email addresses have a higher probability
to be marked as spam emails (because they are not sent through an
authenticated connection of the “From” email address);
if the person receiving the email replies to it, the reply is going
to be sent to the other email addresses that was specified in Outlook.
So make sure you actually have access to this email address, otherwise
you will not get to see the email replies;
the person receiving the email can see that the email was sent through a different email account. As you can see below (the photo is in the link above), Gmail shows that the email was sent through a different domain: Email sent from other address.
As a final comment, don’t think about using the “Send from other email
address” feature as a tool to send spam! Obviously, you shouldn’t
think to send spam emails anyway, but this Outlook 2013 feature
actually uses the SMTP email server of your Outlook account, meaning
that – if your email is marked as spam – it can be tracked back to
your actual email account / SMTP server.
For Thunderbird you can do this by configuring multiple identities. Here is how: Using Identities in Thunderbird
