How can I immediately target all my existing LinkedIn connections with a specific Campaign?

First you will need to create a Lead List based on either a Sales Navigator Saved Search or a LinkedIn search URL that is filtered for 1st connections (and any other filters you wish to add to the segment).

You can now use that Lead List in a new Campaign. For example if you wanted to send them a direct message simply select the Send DM/InMail task type and add the message you wish to send.

Finally launch the Campaign to queue the messages to be sent.