The most straightforward way is to set up a system filter and include a command such as:

unseen deliver mailbox@whatever.domain

This sends a copy of every message to mailbox@whatever.domain (unless the message already has that recipient - Exim never does duplicate deliveries). To save only outgoing messages, you need to come up with a definition of what outgoing means. Typically, this might be a check on the sender address and/or on the originating host. Here is an example:

if $sender_address_domain is mydomain.com and
   ${mask:$sender_host_address/24} is 192.168.124.0/24
then
  unseen deliver mailbox@whatever.domain
endif