To forward emails from your Exchange 2010 domain to the external email address, first you have to create mail enabled contact.

Open Exchange Management Console and navigate to Recipient Configuration -> Mail Contact.

Right click in the middle pane and create New Mail Contact…

Select New Contact to create new contact or Existing Contact to mail enable the existing one.

On the next screen specify the OU (Organizational Unit in AD) if required, enter contact’s details and email address.

Finish the wizard to complete the process.

Now you have to create a mailbox and configure Exchange to forward emails to the mail enabled contact.

In Exchange Management Console navigate to Recipient Configuration -> Mailbox and in the right pane click New Mailbox… to start the wizard.

Select User Mailbox on the wizard screen.

On the next screen select New User, click next and enter user details. Click Next.

Enter mailbox alias (name in front of @yourdomain.com in address). Click Next.

Finish the wizard.

In Exchange Management Console, Recipient Configuration -> Mailbox find newly created mailbox, right click on it and select Properties.

Go to Mail Flow Settings tab select Delivery Options and click Properties button.

Enable Forward To and select mail enabled contact created earlier.